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OALIFOENIA  STATE   SERIES 


ADVANCED 

AKITHMETIC 


BY 


J.  W.  McCLYMONDS  and  D.  R,  JONES 
I  ( 


EBVI8KD    BY   THE 

STATE  TEXT-BOOK  COMMITTEE 

ANI>   APPROVED    BT   THE 

STATE  BOAKD  OF  EDUCATION 


SACRAMENTO 
Friend  Wm.  Richardson,  Supt.  State  Printing. 


Copyright,  1910,  by 
THE  PEOPLE  OF  THE  STATE  OF  CALIFORNIA. 


Copyright,  1907,  by 
J.  W.   McCLYMONDS  and  J).   E.  JONES. 


lyt 


iiiUi^^'nim  .i?^^ 


Jn  the  compilation  of  thin  book  certain  matter  from 
an  "  EmentiaU  of  Arithmetic''''  by  J.  W.  McClymondt 
and  I).  It.  Jones  has  been  used.  All  such  matter  it 
protected    by  the   copyright  entries  noted  above. 


12th  Ed.— 25.000— Oct.,  1914. 


PREFACE 

This  text  is  designed  for  use  in  the  grammar  grades, 
following  the  completion  of  the  Elementary  Arithmetic 
of  the  same  series.  In  the  preparation  of  this  text  the 
authors  have  aimed  (a)  to  secure  skill  in  numerical  com- 
putations and  (6)  to  develop  the  power  necessary  to  the 
solution  of  any  practical  problem  that  may  arise  in  the 
common  experiences  of  life. 

The  following  are  some  of  the  distinguishing  features 
of  this  text: 

1.  The  text  contains  an  unusually  large  number  of 
exercises  that  are  designed  to  give  facility  in  numerical 
computations. 

2.  In  the  presentation  of  each  topic  an  effort  has  been 
made  to  stimulate  thought  and  to  develop  self-reliance  on 
the  part  of  the  pupils.  Whenever  the  nature  of  the  work 
admits,  it  calls  for  action  on  the  part  of  the  pupils,  as  in 
making  measurements,  engaging  in  business  relations  with 
others  in  the  class,  etc. 

3.  The  scope  of  the  work  is  restricted  to  the  needs  of 
the  majority  of  persons  in  the  common  experiences  of  life. 
Traditional  materials  that  make  no  contribution  to  the 
mastery  of  the  essentials  of  arithmetic  have  been  carefully 
eliminated.  All  of  the  work  prescribed  in  the  text  proper 
is  easily  within  the  capacity  of  pupils  in  the  grammar 
grades.  Certain  topics  that  are  prescribed  in  some  courses 
of  study  but  purposely  ^j^yo^^fipm  other  courses  have 


4  PREFACE 

been  presented  in  an  Appendix,  so  that  they  may  be  used 
or  omitted,  as  desired  in  each  case,  without  destroying  the 
continuity  of  the  other  work. 

4.  The  problems  of  the  text  have  been  drawn  from  the 
common  field  of  everyday  experience.  The  necessary 
arithmetical  training  is  had  from  dealing  with  practical 
problems  within  the  experience  of  the  pupils.  No  unreal 
problems,  or  problems  dealing  with  artificial  situations, 
or  problems  treating  of  situations  remote  from  the  experi- 
ences of  the  average  pupil  in  the  grammar  grades,  are 
introduced.     The  text  aims  to  teach  arithmetic  only. 

5.  The  text  contains  an  unusual  amount  of  oral  work, 
including  oral  problems  under  every  topic  treated.  The 
oral  problems  are  everywhere  related  to  the  written  work. 
No  additional  text  in  "  mental "  arithmetic  need  be  used 
m  conjunction  with  this  text. 

6.  The  methods  of  the  text  are  those  commonly  em- 
ployed in  business  life. 

7.  The  work  in  fractions  and  compound  numbers  is 
limited  to  the  practical  needs  of  life.  Special  attention  is 
given  in  fractions  to  the  use  of  those  fractions  which 
pupils  must  handle  later  on  as  the  fractional  equivalents 
of  certain  per  cents.  Commission,  Taxes,  Insurance,  etc., 
are  made  part  of  the  work  in  Percentage  and  are  not 
treated  as  separate  topics.  The  work  in  Interest  has  been 
considerably  reduced,  and  but  one  method  of  finding 
interest  is  recommended. 

8.  A  constant  review  of  all  previous  work  is  maintained 
throughout  the  text. 

Finally,  the  aim  of  the  authors  has  been  to  present 
a  course  in  arithmetic  that  will  secure  a  thorough  knowl- 
edge of  the  essentials  of  this  subject. 


CONTENTS 


PART  I 
Review  of  Integers  and  Decimals 

PAGES 

The  Decimal  System  —  Notation  and  Nuix  sration  —  Addition  — 
Subtraction  —  Multiplication  —  Bills  and  Accounts — Divi- 
sion by  Measurement  and  Partitio  i  —  Comparison  —  Meas- 
urements—  Divisibility  of  Numbers 7-89 


PART   II 

Fractions 

Objective  Fractions  —  Ratio  —  Reduction  —  Addition  —  Sub- 
traction —  Multiplication  —  Division  —  Scale  Drawing  — 
Aliquot  Parts  —  Measurements 90-165 

PART   III 

Percentage 

Percentage  —  Profit  and  Loss  —  Commission  —  Insurance  — 
Taxes  —  Customs  and  Duties  —  Trade  Discount  —  Interest 
—  Promissory  Notes  —  Partial  Payments  —  Compound  Inter- 
est—Bank Discount— Present  Worth        .        .        .      166-220 


PART   IV 

Form^  and  Measurements 

Lines  —  Angles  —  Surfaces  —  Solids  —  Longitude^  and   Time  — 

Ratio r       .        .     221-243 

5 


6  CONTENTS 

PART   V 
Powers  and  Roots 

PAOX8 

Powers  —  Square  Root — Right-angled  Triangles — Similar  Sur- 
faces and  Solids 244-255 

PART  YI 
Appendix 

Corporations,  Stocks,  and  Bonds  —  Commission  and  Brokerage 
—  Trade  Discount  —  Partial  Payments  —  Interest  Table - 
Exact  Interest  —  State  and  Local  Taxes  —  Customs  and 
Internal  Revenue  —  Banking  —  Life  Insurance  —  The  Equa- 
tion—  Proportion  —  Surfaces  and  Solids  —  Measurement  of 
Public  Lands  —  IMetric  System  —  Tables  of  Denominate 
Measures  —  Table  of  Compound  Interest     .        .        .      256-320 

Index 321-324 


ESSENTIALS   OF  ARITIJMETiq,:^ 

PART    I  ..^i'  -'^'-'•'' 

REVIEW   OF   INTEGERS    AND   DECIMALS 

1,   The  Decimal  System. 

1.  A  unit  is  a  single  thing,  or  a  group  of  things 
regarded  as  a  single  thing,  as  a  book,  an  apple,  a  box  of 
apples,  etc.  A  unit  is  represented  by  the  least  whole 
number,  one  (1). 

2.  Point  to  several  units  of  the  same  thing  in  your 
schoolroom.  Can  you  think  of  a  way  by  which  you  could 
tell  your  parents  how  many  children  there  are  in  your 
room  without  using  number? 

3.  Any  definite  quantity  used  to  measure  quantity  of 
the  same  kind  is  called  a  unit  of  measure. 

The  unit  of  6  is  1 ;  of  6  cows  is  1  cow ;  of  9  ft.  is  1  ft. 
The  inch,  foot,  yard,  rod,  and  mile  are  units  used  to  meas- 
ure length  or  distance.  Name  the  units  used  to  measure 
areas.  What  is  the  unit  of  10?  of  1^10?  In  finding  the 
number  of  hats  at  $2  each  that  can  be  bought  for  $10,  the 
unit  of  measure  is  82.  What  is  the  unit  of  measure  in 
finding  the  number  of  4-ft.  shelves  that  can  be  made  from 
a  board  12  ft.  long? 

4.  Name  the  units  used  to  measure  liquids;  time; 
weight. 


8 


REVIEW  OF   INTEGERS  AND  DECIMALS 


5.  Ih  the  number  111,  the  1  at  the  right  denotes  some 
unit,  and  the  1  next  toward  the  left  denotes  a  unit  ten  times 
as  great,  and  the  1  at  the  left  denotes  a  unit  ten  times  the 
second  unit,  or  one  hundred  times  the  first  unit.  This 
aaay-be  shp\Yij  thus : 


one  hundreds'  uhit 


one  tens'  unit 


one  unit 


100 


10 


6.  In  236,  the  6  represents  6  units ;  the  3  represents 
3  units,  each  of  which  is  ten  times  each  of  the  units  repre- 
sented by  6 ;  and  the  2  represents  2  units,  each  of  which 
is  ten  times  each  of  the  units  represented  by  3,  or  one 
hundred  times  each  of  the  units  represented  by  6. 

7.  Tell  what  each  figure  represents  in  125,  47,  352. 

8.  In  30,  the  0  shows  that  there  are  no  units  of  ones ; 
and  the  3  represents  3  units  of  tens.  What  does  each  figure 
represent  in  60,  600,  405,  530,  203,  478,  700,  520? 

9.  In  324,  the  units  represented  by  4  are  called  units 
of  the  first  orders  or  of  unM  order  ;  the  units  represented 
by  2  are  called  units  of  the  second  order^  or  of  tens'*  order  ; 
and  the  units  represented  by  3  are  called  units  of  the 
third  order ^  or  of  hundreds*  order, 

10.  Our  number  system  is  a  decimal  system.  Decimal 
means  tens,  A  decimal  system  is  one  in  which  ten  units  of 
one  order  are  equal  to  one  unit  of  the  next  higher  order. 

The  decimal  system  is  believed  to  have  had  its  origin  in  the  prac- 
tice of  using  the  fingers  for  counting. 


DECIMAL  SYSTEM  9 

11.  Beginning  at  the  left  of  111,  the  1  in  the  third 
order  represents  some  unit ;  the  1  in  the  second  order 
represents  a  unit  one  tenth  as  great ;  and  the  1  in  the 
first  order  represents  a  unit  one  tenth  as  great  as  that 
represented  by  a  unit  of  the  second  order.  A  unit  one 
tenth  as  great  as  that  represented  by  the  1  in  the  first 
order  may  be  represented  by  1  written  to  the  right  of  a 
decimal  point  (.)  placed  to  the  right  of  units'  order,  thus  : 
.1  (111.1).  A  unit  one  tenth  as  great  as  this  last  unit 
may  be  represented  by  1  written  in  the  second  place  to 
the  right  of  the  decimal  point,  thus :   .01  (111.11). 

12.  .1  is  read  one  tenth;  .01  is  read  one  hundredth ;  .11 
is  read  eleven  hundredths ;  1.1  is  read  one  and  one  tenth; 
A  is  read/(92^r  tenths.     Read  6.7;  8.05;  56.25. 

13.  The  decimal  point  is  placed  after  the  figure  that 
represents  whole  units.  The  figures  to  the  right  of  the 
decimal  point  represent  decimal  parts  of  units.  The  parts 
thus  represented  are  tenths,  hundredtlis,  thousandths, 
etc. ;  and  are  called  decimals. 

14.  A  whole  number  is  called  an  integer.  Write  an 
integer.  On  which  side  of  the  decimal  point  are  integers 
written  ? 

15.  What  is  the  meaning  of  the  word  decimal?  Why 
is  our  number  system  called  a  decimal  system? 

16.  What  does  each  2  in  222.222  represent? 

17.  Write  the  following  so  that  units  of  the  same  order 
are  below  one  another:  45.5,  214.25,  347,  4.315,  17. 

18.  Compare  the  value  of  2  in  24  with  the  value  of  2 
in  240  ;  with  the  value  of  2  in  .24. 

19.  Is  the  system  of  United  States  money  a  decimal 
system?     Explain  your  answer. 


10  REVIEW  OF  INTEGERS   AND  DECIMALS 

NOTATION  AND  NUMERATION  OF  INTEGERS  AND 
DECIMALS 

2.  1.  Numbers  are  commonly  expressed  by  means  of 
figures  (or  digits)  as  5,  10,  etc. ;  by  means  of  words,  a,sfive, 
ten,  etc. ;  and  by  means  of  letters,  as  V,  X,  etc.  The  art 
of  writing  numbers  by  means  of  symbols  is  called  notation. 

The  word  digit  means  finger.     Why  were  the  figures  called  digits  ? 

2.  The  figures  1,  2,  3,  4, 5,  6,  7,  8,  9,  0,  are  called  Arabic 
numerals,  as  they  were  introduced  into  Europe  by  the 
Arabs,  who  borrowed  most  of  them  from  the  Hindus. 
The  system  of  denoting  numbers  by  means  of  figures  is 
called  Arabic  notation. 

3.  The  figure  0  is  called  naught,  cipher,  or  zero.  It  has 
no  value.  It  is  used  to  fill  out  places  that  are  not  oc- 
cupied by  other  figures.  Using  figures,  write  six;  six 
tens  ;    six  hundreds. 

4.  The  art  of  reading  numbers  is  called  numeration. 

5.  Integers  of  more  than  three  places  are  read  more 
easily  when  the  figures  are  separated  by  commas  into 
groups  of  three  each,  beginning  at  the  right.  The  groups 
are  called  periods,  and  each  period  is  named  after  the 
order  of  the  right-hand  figure  in  the  group. 

6.  The  names  of  the  first  four  periods,  and  the  orders 
in  each,  are  as  follows: 


'S 

-s 

% 

•s 

Oedbes : 

1 

S 
^ 

1 

CO 

a 

a 

Tens 
Units 

s 

Tens 
Units 

7_ 

8 

9, 

6 

0 

5, 

8 

4     2, 

0 

3      1 

Peeiods  : 

Billions 

Millions 

Thousands 

Units 

a 

b 

c 

1. 

3,625* 

35,205 

825,380 

2. 

8,017 

20,007 

308,016 

3. 

9,008 

45,500 

950,025 

4. 

6,303 

12,012 

404,040 

5. 

What  is 

the 

)  name  of  the  first  perio( 

.    NOTATION  AND  NUMERATION  11 

3.  Reading  Integers. 

To  read  an  iiiteger  of  more  than  three  figures^  begin  at  the 
right  of  the  number  and  point  off  periods  of  three  places  each. 
Read  the  part  occupying  the  left-hand  period  as  though  it 
stood  alone^  and  add  the  name  of  the  period  ;  then  read  the 
part  occupying  the  next  period  as  though  it  stood  alone,  and 
add  the  name  of  the  period.  Continue  until  units'  period  is 
reached  ;  there  omit  the  name  of  the  period. 

Read  the  following  : 

d 
7,125,380 
6,000,150 
8,040,075 
5,505,050 
?  of  the  second 
period  ?   of  the  third  period  ?    of  the  fourth  period  ? 

6.  How  many  periods  are  there  in  the  numbers  in 
column  a?  hi  c?  d? 

7.  How  many  places  do  the  numbers  in  column  c 
occupy  ?   in  column  d  ? 

8.  What  is  the  name  of  the  left-hand  period  in  the 
numbers  in  column  a? 

9.  Kead  the  left-hand  period  of  the  first  number  in 
column  d.  Read  the  middle  period  of  the  same  number. 
Read  the  number. 

10.  Read  the  third  period  of  the  fourth  number  in 
column  d.     Read  the  second  period.     Read  the  number. 

11.  When  a  number  consists  of  three  periods,  how 
many  places  must  there  be  in  the  lirst  period  ?  in  the 
second?  How  many  places  may  there  be  in  the  third  period  ? 

*  625  is  read  six  hundred  twenty-five. 


12  REVIEW  OF  INTEGERS  AND  DECIMALS 

4.  Writing  Integers. 

To  write  a  number  in  figures^  begin  with  the  higheg, 
period  and  write  it  as  though  it  stood  alone^  and  add  a 
comma;  then  write  the  next  highest  period  as  though  it 
stood  alone,  and  add  a  comma;  continue  until  units'  period 
lias  been  written,  thus:   5,807,050. 

Write  in  figures : 

1.  Three  thousand,  two  hundred  four. 

2.  One  hundred  two  thousand,  eight  hundred  ninety. 

3.  Twelve  million,  eight  hundred  seven  thousand, 
eighty-four. 

4.  Seven  hundred  two  million,  sixteen  thousand. 

5.  Write  numbers  dictated  by  your  teacher. 

Read  aloud  the  following  statements  : 

6.  The  area  of  Rhode  Island  is  1,250  sq.  mi. ;  of  Mas- 
sachusetts is  8,315  sq.  mi.;  of  Illinois  is  56,650  sq.  mi.;  of 
California  is  158,360  sq.  mi. ;  of  Texas  is  265,780  sq.  mi. 

7.  In  1900  the  population  of  Rhode  Island  was  428,556  ; 
of  Massachusetts  was  2,805,346 ;  of  Illinois  was  4,821,550 ; 
of  California  was  1,485,053;  of  Texas  was  3,048,710. 

8.  The  total  number  of  votes  cast  for  president  in  1900 
was  13,964,812.  The  five  states  polling  the  largest  num- 
ber of  votes  were  :  New  York,  1,548,042 ;  Pennsylvania, 
1,173,210;  Illinois,  1,131,894;  Ohio,  1,040,073,  and  Mis- 
souri, 683,656. 

9.  The  grain  production  of  the  United  States  in  1902 
in  measured  bushels  was  as  follows:  Indian  corn, 
2,523,648,312;  wheat,  670,063,008;  oats,  987,842,712; 
barley,  134,954,023 ;  rye,  83,630,592 ;  buckwheat, 
14,629,770. 


ROMAN   NOTATION  18 

ROMAN  NOTATION 
5,   1.    The  letters  used   in   Roman  notation  are: 
I        V         X  L  C  DM 

1         5         10         50         100         500         1000 

2.  The  above  letters  are  called  Roman  numerals. 
Other  numbers  are  represented  by  combinations,  thus : 

a.  Repeating  a  numeral  repeats  its  value.  XXX  de- 
notes 30,  CCC  denotes  300.  The  numerals  V,  L,  and  D 
are  not  repeated.     Why  ? 

h.  If  a  numeral  is  followed  by  another  of  less  value, 
the  sum  of  their  values  is  denoted.  XXVI  denotes  the 
sum  of  10,  10,  5,  and  1. 

c.  If  a  numeral  is  followed  by  another  of  greater  value, 
the  difference  of  their  values  is  denoted.  XC  denotes  the 
difference  of  10  and  100,  or  90;  CD  is  500-100,  or  400. 

d.  A  bar  glaced  over  a  numeral  increases  its  value 
1000  times.     V  denotes  5000 ;  IX  denotes  9000. 

3.  Read  the  following  and  tell  which  of  the  above 
rules  is  illustrated  in  each:  LX,  XL,  CIX,  MDCC,  IV, 
MDCCCCVI,  LXXIV,  MMDXL,  MDCXX,  XLII. 

4.  Write  1776  in  Roman  numerals. 

Model  :  1776  may  be  divided  into  the  parts  1000  —  700  —  70  —  6. 
These  parts  expressed  in  order,  beginning  at  the  left,  are  M  —  DCC  — 
LXX  —  VI.     1776  is  written  MDCCLXXYI. 

5.  Write  in  Roman  numerals :  18,  27,  68,  1492,  1907. 

6.  Write  in  Arabic  figures  XCVI,  XLVII,  XIX, 
LXXIV,   MDCCCXII. 

Roman  numerals  are  frequently  used  to  designate  chapter  numbers 
in  books,  the  hours  on  the  clock  face,  dates  on  monuments  and  pub- 
lic buildings,  etc.  The  M  is  used  to  designate  a  thousand  feet  of 
lumber. 


14  REVIEW   OF   INTEGERS  AND  DECIMALS     • 

UNITED  STATES   MONEY 

6.  1.  The  units  of  United  States  money  are  decimal 
units.  The  standard  unit  of  value  is  the  dollar.  The 
other  units  are  derived  from  it.  The  dime  is  one  tenth  part 
of  the  dollar,  and  the  units  that  represent  dimes  are  there- 
fore written  in  the  first  place  to  the  right  of  the  decimal 
point.  The  cent  is  one  hundredth  part  of  the  dollar,  and 
the  units  that  represent  cents  are  therefore  written  in  the 
second  place  to  the  right  of  the  decimal  point. 

2.  Dimes  are  written  as  cents.  Two  dollars  and  four 
dimes  is  written  thus:  12.40.  This  is  read  two  dollars 
and  forty  cents. 

3.  The  unit  one  dollar  is  written  $1.  The  unit  one 
dime  is  written  $.10.  The  unit  one  cent  is  written  $.01. 
The  unit  one  mill  is  written  f.OOl. 


7.    Reading  United  States  Money. 

Read  the  following 

: 

1.    1425.15 

5.    $30,755 

9. 

$8340.05 

2.    $301.08 

6.   $  7.057 

10. 

$9015.807 

3.    1220.20 

7.   $10,105 

11. 

$7200.50 

4.    $100.10 

8.   $  4.005 

12. 

$1306.065 

8.  Writing  United  States  Money. 
Write  the  following  in  columns  : 

1.  Six  dollars  and  seventy-five  cents. 

2.  Twenty-five  dollars  and  fifty  cents. 

3.  Eighty-five  dollars  and  six  cents. 

4.  Three  hundred  forty  dollars  and  eighty  cents. 

5.  One  hundred  dollars  and  fifty-two  cents. 

6.  Eight  cents.     7.    Thirty-five  cents  and  eight  mills. 


READING   AND   WRITING  DECIMALS  16 

9.  Reading  Decimals.  | 


•a 

00  +3        -^  w         "1"         2 

Okders:     '^wiS         :S'«s^'«.2 

9     13.452     876 

Integers  Decimals 

1.  Memorize  the  number  of  decimal  places  required 
for  each  of  the  first  six  orders. 

Tenths  (first)  .         .'        .         .        .        .         .         .         .5 

Hundredths  (second) 45 

Thousandths  (third) 367 

Ten-thousandths  (fourth) 6745 

Hundred-thousandths  (fifth) 62789 

Million  ths  (sixth) 346329 

To  read  a  decimal,  read  the  number  without  reference  to 
the  decimal  point,  and  add  the  name  of  the  order  of  the 
right-hand  figure. 

2.  .375  is  read  three  hundred  seventy-five  thousandths. 
3.08  is  read  three  and  eight  hundredths.  Read:  .125,  .875, 
4.625,  37.075,  670.005,  3.1416,  2150.42,  .7854. 

10.  Writing  Decimals. 

1.  Write  sixty-two  thousandths.  As  thousandths  is 
the  name  of  the  third  order  to  the  right  of  the  decimal 
point,  three  figures  will  be  required  in  writing  the  num- 
ber. Two  figures  are  necessary  to  denote  sixty-two ;  so 
one  cipher  must  be  supplied.  To  write  sixty-two  thou- 
sandths, first  write  the  decimal  point,  then  write  0,  and 
then    write  62  (.062). 

2.  Write  the  following:  Sixty-nine  ten-thousandths^ 
forty-eight  hundred-thousandths ;  thirteen  thousandths. 


16  REVIEW  OF   INTEGERS  AND   DECIMALS 

ADDITION  OF  INTEGERS  AND  DECIMALS 

11,  1.  A  number  that  is  not  applied  to  any  particular 
thing,  as  6,  43,  etc.,  is  called  an  abstract  number. 

2.  A  number  that  is  applied  to  some  particular  thing, 
as  6  ft.,  43  lb.,  etc.,  is  called  a  concrete  number. 

3.  Quantities  that  are  expressed  in  the  same  unit  of 
measure,  as  3  lb.  and  6  lb.,  are  called  like  quantities. 

4.  Quantities  that  are  expressed  in  different  units  of 
measure,  as  5  lb.  and  4  hr.,  are  called  unlike  quantities. 

5.  Write  two  abstract  numbers;  two  concrete  num- 
bers; two  unlike  quantities.  Like  quantities  can  be 
combined  and  expressed  as  a  single  quantity.  3  ft.  and 
2  ft.  may  be  combined  and  expressed  as  5  ft.  Can  the 
unlike  quantities  5  lb.  and  4  hr.  be  combined  and  ex- 
pressed as  a  single  quantity? 

6.  Units  of  the  same  order  may  be  combined  and 
expressed  as  single  numbers.  3  tens  and  2  tens  are 
5  tens. 

7.  When  two  or  more  numbers  are  combined  and  ex- 
pressed as  a  single  number,  this  number  is  called  their 
sum,  or  amount. 

8.  The  process  of  finding  the  sum  of  two  or  more 
numbers  is  called  addition.  The  numbers  that  are  added 
are  called  addends. 

9.  The  sign  of  addition  is  +  and  is  read  plus. 

10.  This  sign  =  is  the  sign  of  equality,  and  when 
placed  between  two  numbers  is  read  equals  or  is  equal  to^ 
thus:  6  =  4  +  2  means  that  6  is  equal  to  the  sum  of  4 
and  2. 


ADDITION  17 

12.  Oral  Exercises.* 

To  each  number  in  Exs.  1-4,  add  in  succession  3,  2,  7, 
6,  9,  4,  8,  5. 

1.  23  35  84  69  26  88  82  57  47  60 

2.  39  76  48  87  65  74  33  22  81  30 

3.  52  86  49  61  73  95  40  18  67  94 

4.  90  66  38  17  41  93  55  74  12  99 

13.  Add  each  column  as  written.  Add  each  column, 
increasing  the  number  at  the  bottom  of  the  column  by  10, 
by  20,  etc.,  to  90 ;  thus  for  column  a,  having  increased 
the  number  at  the  bottom  of  the  column  by  20:  22,  25, 
29,  etc. 

a       b       c        defghijklmn 
98958797778896 


3 

7 

8 

8 

7 

9 

3 

3 

7 

9 

9 

9 

5 

8 

6 

5 

9 

2 

5 

7 

9 

6 

8 

8 

9 

5 

6 

4 

4 

6 

3 

7 

4 

6 

5 

8 

6 

6 

5 

9 

4 

9 

3 

8 

9 

6 

8 

9 

5 

2 

4 

9 

1 

8 

8 

8 

8 

9 

7 

5 

9 

6 

8 

7 

8 

7 

7 

7 

3 

9 

7 

7 

6 

4 

5 

8 

5 

5 

6 

8 

1 

9 

4 

2 

7 

7 

3 

5 

7 

6 

9 

9 

4 

4 

8 

7 

3 

7 

2 

4 

5 

9 

8 

8 

7 

8 

5 

9 

4 

6 

9 

6 

3 

7 

2 

3 

7 

9 

6 

5 

8 

8 

2 

8 

5 

5 

4 

3 

9 

5 

6 

4 

9 

7 

9 

7 

6 

3 

7 

7 

3 

8 

9 

2 

9 

6 

6 

7 

8 

5 

8 

2 

7 

7 

2 

6 

8 

4 

4 

5 

7 

9 

6 

2 

8 

9 

7 

4 

*  If  the  pupils  require  a  more  extended  drill  upon  addition  than  is  pro- 
vided in  the  above  exercises,  the  method  indicated  in  the  elementary  text 
should  be  followed. 

AR.  — 2 


18  REVIEW  OF   INTEGERS  AND   DECIMALS 

14.  Written  Exercises. 

Numbers  to  be  added  or  subtracted  must  be  written 
so  that  units  of  the  same  order  are  directly  below  one 
another,  units  under  units,  tens  under  tens,  and  tenths 
under  tenths,  etc.     Why  ? 

When  numbers  are  written  so  that  the  decimal  points  are 
directly  below  one  another,  units  of  the  same  order  are 
directly  below  one  another.     Explain. 

Add: 

1.  2.  3. 

$   345.67  $  58.06  $     9.045 

84.075  275.936  590. 

650.  83.07  5.15 

70.004  342.457  69.075 

572.806  34.08  610.75 

6.605  8.125  57.246 

852.451  64.       .  540.375 

^  ^  W  "  Kead  aiduii  6ach  of  ihe  aqove.'  ^ 

7.  Write  the  above  from  dictation. 

8.  Add  74.06  mi.,  6.8  mi.,  320.45  mi.,  17.04  mi. 

9.  Add  64.5  A.,  79.14  A.,  160.75  A.,  321.15  A. 

10.  Add  60.5  cu.  in.,  352.24  cu.  in.,  80.125  cu.  in. 

11.  Add  168.05,  1107.98,  1730.04,  f  9.75,  1894,  $80, 
1740.40,  1375.15,  f486.75,  1836.95,  i.95. 

12.  Add  six  and  nine  hundredths,  thirty-seven  and 
six  tenths,  eighty-five  thousandths,  seven  hundredths. 

13.  Find  the  sum  of  nine  hundred  eighty  and  five 
tenths,  seventy  and  seven  hundredths,  one  hundred  and 
five  thousandths,  six  hundred  twenty-five. 

14.  Write  five  addition  exercises  similar  to  Exs.  1-5 
above  and  add  each.     Read  each  answer. 


4. 

5. 

$405.27 

$  68. 

73.435 

125.87 

487.50 

45.369 

50.258 

845.075 

250.50 

8.75 

.375 

100. 

62.50 

58.268 

ADDlTIOiSr  19 


15. 

Oral  Exercises 

. 

Add 

a 

h 

c 

d 

e 

/ 

9 

h 

i 

1.  40 

60 

130 

120 

60 

150 

90 

140 

80 

50 

20 

90 

30 

140 

40 

30 

60 

20 

2.  TO 

40 

140 

50 

60 

80 

70 

20 

70 

25 

54 

63 

139 

42 

59 

96 

192 

.36 

3.  29 

55 

79 

56 

54 

89 

46 

92 

m 

90 

80 

70 

90 

80 

70 

80 

90 

60 

4.  23* 

43 

64 

36 

59 

54 

68 

94 

39 

89 

52 

95  . 

94 

43 

46 

43 

36 

27 

5.  Frank  weighs  95  lb.  and  his  little  brother  weighs 
34  lb.     How  much  do  they  both  together  weigh  ? 

6.  A  farmer  has  46  sheep  and  his  neighbor  has  54 
sheep.     How  many  have  both  together? 

7.  A  man  paid  f  94  for  a  wagon  and  836  for  a  harness. 
How  much  did  both  cost  him  ? 

8.  Mr.  White  had  23  head  of  cattle  and  bought  39 
more.     How  many  had  he  then  ? 

9.  A  girl  spent  50  ^  for  cloth  and  45/  for  lace.     How 
much  did  she  spend  for  both  ? 

10.  A  boy  placed  60/  into  his  bank  one  week  and  46/ 
the  next  week.  How  much  did  he  put  into  the  bank  in 
the  two  weeks? 

11.  A  girl  spent  20/  for  stamps,  25/  for  some  meat, 
and  50  /  for  sugar.     How  much  did  she  spend  for  all  ? 

12.  Make  and  solve  ten  oral  problems  in  addition. 

♦  Add :  89,  109,  112. 


20  REVIEW  OF  INTEGERS  AND  DECIMALS 

SUBTRACTION  OF  INTEGERS  AND  DECIMALS 

16.  1.  Like  quantities,  such  as  5  marbles  and  9  marbles, 
may  be  compared,  and  the  difference  between  them  found, 
thus: 

9  marbles:        ••••••••• 

5  marbles:        •      •      •      •      • 

2.  If  there  is  added  to  5  marbles  a  quantity  that  will 
make  it  equal  to  9  marbles,  how  much  is  added  ?  This 
amount  is  the  difference  between  the  two  quantities. 

3.  If  that  part  of  9  marbles  that  is  equal  to  5  marbles 
is  taken  from  9  marbles,  liow  many  will  remain  ?  This 
remainder  is  the  difference  between  the  two  quantities. 

4.  How  does  the  difference  as  found  in  Ex.  3  compare 
with  the  difference  as  found  in  Ex.  2  ? 

5.  The  difference  between  the  two  quantities  may  be 
found  by  answering  either  of  the  following  questions: 

a,    5  marbles  and  how  many  marbles  are  9  marbles  ? 
h.    5  marbles  from  9  marbles  leaves  how  many  marbles  ? 
In  either  case,  the  answer  is  known  by  recalling  that 
the  sum  of  5  marbles  and  4  marbles  is  9  marbles. 

6.  The  difference  between  two  numbers  is  the  number 
which  when  added  to  one  number  makes  the  other  number. 

7.  The  process  of  finding  the  difference  between  two 
numbers  is  called  subtraction. 

8.  The  number  to  which  the  difference  is  added  is 
called  the  subtrahend. 

9.  The  sum  of  the  subtrahend  and  difference  is  called 
the  minuend. 

Or  the  subtrahend  is  the  number  which  is  subtracted,  and  the 
minuend  is  the  number  from  which  the  subtrahend  is  taken. 


SUBTRACTION  21 

17.  Oral  Exercises. 

a  b  c 

1.  6  and  —  are  11       8  and  —  are  12       9  and  —  are  16 

2.  9  and  —  are  14  7  and  —  are  13  8  and  —  are  14 

3.  8  and  —  are  11  4  and  —  are  11  7  and  —  are  11 

4.  7  and  —  are  12  9  and  —  are  15  5  and  —  are  14 

5.  5  and  —  are  13  5  and  —  are  11  6  and  —  are  15 

6.  4  and  —  are  12  3  and  —  are  12  9  and  —  are  13 

7.  8  and  —  are  15  5  and  —  are  12  8  and  —  are  16 

8.  7  and  —  are  16  6  and  —  are  14  7  and  —  are  14 

9.  9  and  —  are  17  8  and  —  are  13  8  and  —  are  17 

10.  6  and  —  are  12       7  and  —  are  15       9  and  —  are  11 

11.  9  and  —  are  12        9  and  —  are  18       6  and  —  are  13 

9 

12.  _  r  is  read  5  and  how  many  are  9?   Or,  ^  from  9  leaver 

how  many?     Use  the  form  with  which  you  are  familiar. 

13.  The  sign  of  subtraction  is  ~,  and  is  called  minus. 
It  indicates  that  the  number  that  follows  it  is  to  be  sub- 
tracted from  the  number  that  precedes  it.  7  —  4  is  read 
seven  minus  four, 

18.  Explanation  of  Subtraction. 
1.    Find  the  missing  addend. 

(one  addend)  The  other   addend   may  be 

2874  (one  addend)  ^^^"^  ^7  a^d^^S  *°  *^«  g^^^^ 

rnor«   y             £   ^           i  i       i  n  addeiid  the   number   that  will 

ozdo  (sum  01   two  addends)  .     ^,            ^,        ^      j « 

^                                              ^  give  the  sum,  thus  :  4  and  2  are 

6 ;  7  and  6  are  13 ;  carry  1  to  8,  making  it  9  ;  9  and  3  are  12 ;  carry 

1  to  2,  making  it  3;  3  and  2  are  5.     Missing  addend,  2362. 


22  REVIEW  OF  INTEGERS  AND  DECIMALS 

2.    From  5236  subtract  2874. 

Model  a :      5236         -A-fld  to  the  subtrahend  the  number  that  will 

2874     give  the  minuend,  thus :  4  and  2  are  6  ;  7  and 

23(32     6  are  13  ;  carry  1  to  8  as  in  addition,  making  it 

9;  9  and  3  are  12;  carry  1  to  2  as  in  addition, 

making  it  3;  3  and  2  are  5.    AVrite  the  answer  as  in  the  model. 

This  is  known  as  the  Austrian,  or  additive,  method. 

Model  b :  Subtract  thus :  4  from  6  leaves  2 ;  as  7  tens  cannot  be 
taken  from  3  tens,  1  hundred  is  "  borrowed "  from  2  hundreds  and 
called  10  tens;  10  tens  and  3  tens  are  13  tens;  7  tens  from  13  tens 
leaves  6  tens  ;  as  1  hundred  was  borrowed  from  2  hundreds,  there  is  left 
1  hundred;  as  8  hundreds  cannot  be  taken  from  1  hundred,  1  thousand 
is  borrowed  from  5  thousands  and  called  10  hundreds ;  adding  10  hun- 
dreds to  1  hundred  gives  11  hundreds  ;  8  hundreds  from  11  hundreds 
leaves  3  hundreds ;  as  1  thousand  was  taken  from  5  thousands,  there 
are  left  4  thousands;  2  thousands  from  4  thousands  leaves  2  thousands. 

Model  c  :  If  the  same  number  is  added  to  both  the  minuend  and 
the  subtrahend,  the  difference  remains  unchanged.  Subtract  thus: 
4  from  6  leaves  2 ;  as  7  tens  cannot  be  taken  from  3  tens,  add  10  tens 
to  3  tens,  making  13  tens ;  7  tens  from  13  tens  leaves  6  tens  ;  as  10  tens 
were  added  to  the  minuend,  the  same  number  must  be  added  to  the 
subtrahend,  so  1  hundred  (10  tens)  is  added  to  8  hundreds,  making  9 
hundreds;  as  9  hundreds  cannot  be  taken  from  2  hundreds,  10 
hundreds  are  added  to  2  hundreds,  making  12  hundreds;  9  hundreds 
from  12  hundreds  leaves  3  hundreds;  as  10  hundreds  were  added 
to  the  minuend,  the  same  number  must  be  added  to  the  subtrahend, 
so  1  thousand  (10  hundreds)  is  added  to  2  thousands,  making  3 
thousands ;  3  thousands  from  5  thousands  leaves  2  thousands. 

19.  Written  Exercises. 

Solve : 

1.  38,256-21,359  6.  1,106,800-289,060 

2.  40,175-19,688  7.  4,083,453-613,757 

3.  85,430-41,856  8.  3,256,845-465,868 

4.  93,950-17,275  9.  4,741,242-572,847 

5.  97,204-57,240  lO.  2,814,004-935,940 


SUBTRACTION  23 

20.  Oral  Exercises. 
Subtract : 

a          h          c          d         e          f  g           hi 

1.      40       140     150     100     120     150  120       90     110 

20        30      20       40       30       60  90      40       50 




2. 

95 
40 

126 
90 

83 
50 

142 

60 

149 

80 

155 

70 

153 

20 

129 
40 

124 

80 

3. 

124* 
92 

109 
44 

138 
85 

96 
13 

75 
24 

139 
68 

136 
45 

88 
16 

99 
44 

4. 

75 1 

38 

142 

96 

34 
19 

57 
29 

83 
68 

74 

18 

42 

27 

36 
19 

52 

29 

5.  Harry  bought  120  yd.  of  string  and  used  85  yd.  for 
a  kite  string  and  gave  the  rest  to  George.  How  many 
yards  did  he  give  to  George  ? 

6.  A  farmer  had  52  head  of  cattle  and  sold  29.  How- 
many  had  he  left  ? 

7.  Mary  read  87  pages  in  a  book  that  contained  124 
pages.  How  many  more  pages  must  she  read  to  complete 
the  book  ? 

8.  There  are  38  pupils  in  Room  A  and  47  in  Room  B. 
How  many  pupils  are  there  in  both  rooms  ?  How  many 
more  pupils  are  there  in  Room  B  than  in  Room  A? 

9.  The  frontage  of  a  certain  city  lot  is  40  ft.  and  its 
depth  is  135  ft.  Find  the  difference  between  the  depth 
and  frontage  of  the  lot. 

*  Suggestion.     The  difference  iDetween  92  and  124  Is  30  and  2,  or  32. 
t  Suggestion.     The  difference  between  38  and  75  is  80  (38  to  68)  and 
7  (68  to  76),  or  37  ;  or  40  less  3,  or  37. 


24  REVIEW  OF  INTEGERS  AND  DECIMALS 

21,   Before  solving,  represent  each  by  a  diagram. 

1.  Two  boys  started  from  the  same  place.  One  boy 
rode  east  32  mi.  and  the  other  boy  rode  west  24  mi.  How 
far  apart  were  they  then? 


W. 


2^  mi.  S  32  mL 

From  S.  to  E.  is  32  mi.  and  from  S.  to  W.  is  24  mi. 
From  E.  to  W.  is  the  sum  of  32  mi.  and  24  mi.,  or  ^Q  mi. 

2.  Two  boys  started  from  the  same  place.  One  rode 
east  32  mi.  and  the  other  rode  east  24  mi.  How  far  apart 
were  they  then  ? 

3.  How  far  apart  are  two  places,  if  one  is  40  mi.  north 
of  the  center  of  a  certain  city,  and  the  other  is  65  mi. 
south  of  the  center  of  the  same  city  ? 

4.  Mary  lives  8  blocks  east  of  the  schoolhouse,  and 
Ethel  lives  14  blocks  west  of  the  schoolhouse.  How  far 
apart  do  the  girls  live  ? 

5.  Two  trains  left  a  certain  station  at  the  same  time, 
going  in  opposite  directions.  How  far  apart  were  they  at 
the  end  of  2  hours,  if  one  traveled  at  the  average  rate  of 
42  mi.  an  hour,  and  the  other  at  the  average  rate  of  36  mi. 
an  hour  ? 

6.  How  far  apart  would  the  trains  mentioned  in  Prob. 
5  be  at  the  end  of  2  hours,  if  both  traveled  in  the  same 
direction  ? 

7.  In  a  bicycle  race  Frank  and  Henry  rode  around  a 
park  400  ft.  long  and  200  ft.  wide.  When  Frank  had 
ridden  once  around  tlie  park,  Henry  hud  gained  200  ft. 
on  him.  At  tlie  same  rate  of  gain,  how  many  times  will 
Frank  ride  around  the  park  before  Henry  overtakes  him  ? 


SUBTRACTION  25 

22.  United  States  Money. 

Write  units  of  tlie  same  kind  below  one  another.     Do 
not  supply  unnecessary  O's. 

1.  Subtract:     a.  $12.75  from   137.25;    h.  $12   from 
137.25;  c.  112.75  from  $37. 
Model  a:   $37.25     Model  &:   $37.25     Model  c:   $37. 

12.75  12.  12.75 

$21.50  $25.25  $24.25 

Solve : 

2.  $307.57    -$200.69  6.  120.375-93 

3.  $925.07    -$570.80  7.  690.125-209 

4.  $700.40    -$180.05  8.  542-45.78 

5.  $860.455 -$280  9.  640-70.65 

10.  Read  aloud  each  of  the  above  amounts. 

11.  Write  the  above  amounts  from  dictation. 

23.  Decimals. 

Subtract : 
1.  2.  3.  4. 

320.564  450.125  35.7  600. 

206.7  86.75  6.875  57.375 


5.  A  man  owned  158.15  acres  of  land.  He  sold  79.5 
acres.     How  many  acres  had  he  left  ? 

6.  If  it  is  844.7  mi.  from  San  Francisco  to  Ogden  and 
1004.7  mi.  from  Ogden  to  Omaha,  how  far  is  it  from  San 
Francisco  to  Omaha?  How  much  farther  is  it  from  Ogden 
to  Omaha  than  from  San  Francisco  to  Ogden  ? 

7.  A  cubic  foot  of  rain  water  weighs  62.5  lb.  and  a 
cubic  foot  of  petroleum  weighs  54.875  lb.  How  much 
heavier  is  a  cubic  foot  of  rain  water  than  a  cubic  foot  of 
petroleum  (kerosene)  ? 


26  REVIEW  OF  INTEGERS   AND  DECIMALS 

24.  1.  Show  the  effect,  if  any,  upon  the  difference  : 
(a)  of  adding  the  same  number  to  both  minuend  and 
subtrahend;  (6)  of  subtracting  the  same  number  from 
both  minuend  and  subtrahend.  Illustrate  each  with 
several  exercises. 

2.  Write  ten  exercises  in  subtraction  of  decimals  and 
solve  each. 

25.  Oral  Exercises. 

1.  Name  five  combinations  whose  sums  are  10. 

When  these  combinations  occur  in  a  column,  they  should 
be  treated  as  10.     Exercise  a  below  may  be  added :  15, 

25,  32,  42,  48,  58,  66,  16.     Add  the  following  exercises  in 
a  similar  manner  : 

abcdefgh        i       j       k        I      m 

57841574   5   87   89 

5326.9526573   98 

8697648567712 

r5   3841347638   97 

15   726976348246 

68798866766   64 

r5   7269782145  6      b 

l5   384132898   559 

75586997723   21 

r5   3269541379   85 

15   7841569891   79 

66789999977   83 

989786   5   29688^4 

2.  Write  ten  columns,  in  each  of  which  some  of  the 
five  combinations  wliose  sums  are  10  occur  several  times. 
Add  these  columns. 


SUBTRACTION 
26.  Written  Exercises. 


27 


Nbw  England  States 

Area  in 
Sy.  MiLBs 

Gkbat  Lakes 

Area  in 
Sq.  Miles 

33,040 

9,305 
9,565 
8,315 
1,250 
4,990 

Superior 

Huron 

Michigan 

Erie 

Ontario 

31,200 

23,800 

22,450 

9,960 

7,240 

New  Hampshire 
Vermont    .     .     . 
Massachusetts    . 
Rhode  Island     . 
Connecticut   .     . 

1.  Find  the  combined  area  of  the  New  England  states; 
of  the  Great  Lakes. 

2.  Find  the  difference  between  the  combined  area  of 
the  New  England  states  and  of  the  Great  Lakes. 

3.  Compare  the  area  of  Vermont  with  the  combined 
area  of  Massachusetts  and  Rhode  Island. 

4.  Find  the  difference  between  the  area  of  Lake 
Ontario  and  the  combined  area  of  Rhode  Island  and 
Connecticut. 

5.  Find  the  difference  between  the  area  of  Lake  Su- 
perior and  the  combined  area  of  Lakes  Huron,  Erie,  and 
Ontario. 

6.  Compare  the  area  of  Maine  with  the  area  of  Lake 
Superior. 

7.  Find  the  difference  between  the  area  of  Maine  and 
the  combined  area  of  the  other  five  New  England  states. 

8.  The  area  of  Missouri  is  69,415  sq.  mi.  Compare 
the  area  of  Missouri  with  the  combined  area  of  the  New 
England  states. 


28  REVIEW  OF  INTEGERS  AND  DECIMALS 

27.  Written  Exercises. 

In  solving  a  problem,  follow  these  steps  in  the  order 
given  : 

a.    Read  the  problem  carefully^  if  convenient,  aloud. 

h.  State  what  facts  are  given  in  the  problem  and  what 
fact  you  are  asked  to  find. 

c.  Determine  what  relation  the  given  facts  have  to  one 
another,  and  state  what  operation  you  must  use  in  finding 
the  facts  that  are  asked  for,  —  whether  you  must  add  or 
subtract,  etc. 

d.  Make  an  estimate  of  the  answer.  When  you  have 
found  the  answer,  compare  it  with  this  estimate. 

1.  A  man  bought  a  house  for  $2400  and  sold  it  for 
$3000.     Find  the  amount  gained. 

2.  A  farmer  sold  his  farm  for  $7500,  which  was  $1800 
more  than  it  cost  him.  How  much  did  he  pay  for  the 
farm  ? 

3.  A  farmer  bought  a  farm  for  $6250  and  sold  it  at  a 
gain  of  $1200.     How  much  did  he  get  for  the  farm? 

4.  By  selling  a  farm  for  $4500,  a  farmer  received  $900 
less  than  it  cost  him.     How  much  did  it  cost  him? 

5.  A  room  is  24  ft.  long  and  18  ft.  wide.  Find  how 
many  feet  of  picture  molding  it  will  require  for  the  room. 

6.  After  drawing  out  $2300  from  a  bank,  a  merchant 
had  $760  left  in  the  bank.  How  much  had  he  on  deposit 
in  the  bank? 

7.  A  merchant  had  $1600  on  deposit  in  a  bank  on 
Jan.  1,  1907.  On  Jan.  2  he  drew  out  $200.  On  Jan.  5 
he  deposited  $750.  On  Jan.  15  he  drew  out  $2000. 
How  much  had  he  left  in  the  bank? 


SUBTRACTION 


29 


28.  Written  Exercise*. 

1.  The  total  production  of  corn  in  the  United  States  in 
1899  was  2,666,440,279  bu.  In  1889  tiie  total  production 
was  2,122,327,547  bu.  How  much  had  the  production 
increased  during  the  decade  (10  years)? 

2.  In  1899  the  production  of  corn  in  Illinois  was 
398,149,144  bu.  The  production  in  1889  was  289,697,256 
bu.     Find  the  increase  in  production  during  the  decade. 

3.  From  the  amounts  given  in  Frobs.  1  and  2,  find  the 
total  number  of  bushels  produced  in  all  states  other  than 
Illinois  in  1899. 

4.  The  total  production  of  rice  in  the  United  States  in 
1899  was  as  follows : 


Louisiana   . 

South  Carolina 

Hawaii 

Georgia 

North  Carolina 

Texas 

Florida 

Alabama     . 

Mississippi 

Arkansas    . 

Virginia 


172,732,430  lb. 
47,360,128  lb. 
83,442,400  lb. 
11,174,562  lb. 

7,892,580  lb. 

7,186,863  lb. 

2,254,492  lb. 

926,946  lb. 

739,222  lb. 

8,630  lb. 

4,374  lb. 


a.    Read  aloud  the  above  quantities. 
h.    Write  the  above  from  dictation. 

c.  Find  the  total  number  of  pounds  produced. 

d.  Was  the  amount  produced  by  Louisiana  more  or  less 
than  that  produced  by  all  others  combined,  and  how  much  ? 

e.  Compare  the    amount  produced  in   South  Carolina 
with  the  total  amount  produced  in  Hawaii  and  Georgia. 


30  REVIEW  OF  INTEGERS  AND  DECIMALS 

29.  Written  Exercises. 

1.  Ill  1890  the  population  of  San  Francisco  was  298,997, 
and  in  1900  it  was  342,782.  Find  the  increase  in  popula- 
tion between  1890  and  1900. 

2.  In  1898  the  population  of  London  was  4,504,766, 
and'  in  1900  the  population  of  New  York  was  3,437,202 
and  of  Chicago  was  1,698,575.  Find  the  difference  be- 
tween the  population  of  London  and  the  combined  popu- 
lation of  New  York  and  Chicago. 

3.  The  area  of  the  earth's  surface  is  about  196,940,000 
sq.  mi.  Of  this,  141,486,000  sq.  mi.  is  covered  with 
water.     Find  the  area  of  the  land. 

4.  A  cattle  dealer  bought  some  cattle,  for  which  he  paid 
$  380.  He  paid  out  f  67  for  feed  and  care  of  the  cattle. 
He  then  sold  them  for  ef  500.  How  much  was  his  net 
profit,  that  is,  the  profit  after  deducting  all  expenses  ? 

5.  A  real  estate  dealer  bought  a  city  lot  for  11750. 
He  built  a  house  on  it  that  cost  $  3275  and  then  sold  the 
property  for  $  6000.     Find  the  amount  of  his  gain  or  loss. 

6.  The  total  area  under  broom-corn  cultivation  in  the 
United  States  in  1899  was  178,584  acres.  In  1889  it  was 
93,425  acres.  How  much  was  the  increase  in  the  area 
under  cultivation  during  the  decade  ? 

7.  The  appropriation  for  the  maintenance  of  the  navy 
for  1907  was  $  98,773,692,  for  the  military  $  72,305,270,  for 
pensions  $  143,746,106.  How  much  was  appropriated  for 
these  three  purposes  ?  How  much  more  were  the  combined 
appropriations  for  the  navy  and  military  than  for  pensions  ? 

8.  The  total  appropriations  of  the  government  for  1907 
amounted  to  $5  701,551,566.  Find  the  appropriations  for 
all  purposes  other  than  the  navy,  military,  and  pensions. 


MULTIPLICATION  31 

MULTIPLICATION  OF  INTEGERS  AND  DECIMALS 

30,  1.    Find  the  sum  of  a  column  of  four  2's. 

2.  The  sum  of  a  column  of  four  2's  is .     In  this 

column  the  addend  2  is  repeated  4  times.  Four  times  2 
are .  2 

X  4 

3.  Four  2's  are  8  may  be  indicated  thus :      -  •     Here 

2  is  taken  4  times,  or  is  multiplied  hy  4.  The  2  is  the 
addend  that  is  repeated,  and  the  4  tells  the  number  of 
times  this  addend  is  repeated. 

%   82 

4.  The   sum   $32   taken   4    times   may  be     ^   oo 

found  by  addition,  thus:  ^  g2 

It  may  also  be  found  by  ^   ^^ 

multiplication,  thus :  ^Toq  $128 

Since  four  2's  are  8  and  four  3's  are  12,  four  |32's  are  $128. 

5.  Find  the  cost  of  3  cows  at  $48  each  by  addition; 
by  multiplication.     Which  method  is  the  shorter? 

6.  A  man  paid  the  following  amounts  for  three  horses  : 
$120,  $85,  and  $100.  Can  the  cost  of  the  three  horses  be 
found  by  multiplication  ?     Give  a  reason  for  your  answer. 

7.  Multiplication  is  the  process  of  taking  one  number 
as  many  times  as  there  are  units  in  another. 

8.  The  number  that  is  multiplied  is  called  the  multi- 
plicand; the  number  by  which  we  o^^c^oa         i^-  t       j 

u-  1     •        ^^  A  ^x.  1.-  1-  ^284,  multiplicand, 

multiply  IS  called  the  multiplier;  «        if  r 

and  the  result  obtained  is  called  ^^t^         ,     . 

,,  ,     ^  $702,  product, 

the  product. 

^284  .       .        ^^.^. 

9.  Express         «  as   an  exercise   in   addition. 


32  .REVIEW  OF   INTEGERS  AND  DECIMALS 

10.  Regard  the  multiplicand  as  a  repeated  addend  and 
tlie  multiplier  as  the  number  of  times  the  addend  is 
repeated. 

11.  The  product  and  the  multiplicand  are  always  like 
quantities.     Why  ? 

12.  The  multiplier  is  always  an  abstract  number.  It 
tells  how  many  times  the  multiplicand  is  to  be  taken,  or 
how  many  times  the  addend  is  to  be  repeated. 

13.  The  sign  of  multiplication  is  x  .  It  indicates  that 
the  number  before  it  is  to  be  multiplied  by  the  number 
after  it.  $3  x  4  is  read  83  multiplied  by  4.  The  sign 
(  X  )  is  sometimes  used  in  place  of  the  word  times  in  such 
an  expression  as  2  times  f  5. 

14.  Express  each  of  the  following  in  the  form  of  addi- 
tion:  $4x6;  5  lb.  X  3;  4  times  6  yd. ;  7  in.  x  8;  9x4. 

31.  Law  of  Commutation. 
1.  In  the  following  diagram  there  are  3  rows  of 
squares,  with  4  squares  in  each  row.  Or,  there  are  4 
rows  of  squares,  with  3  squares  in  each 
row.  There  are  in  all  12  squares.  We 
see  that  3  times  4  squares  and  4  times 
3  squares  are  the  same  number  of 
squares. 

2.  Find  the  sum  of  three  4's  and  of 
four  3's.  Since  the  sum  of  four  3's  is  the  same  as  the 
sum  of  three  4's,  the  product  of  3  and  4  is  the  same, 
without  regard  to  which  is  multiplier  and  which  is  mul- 
tiplicand. 

3.  Show  by  the  addition  of  columns  that  the  sum  of 
five  6*s  equals  the  sum  of  six  5's.  Show  by  a  diagram 
that  5  times  6  squares  equals  6  times  5  squares. 


MULTIPLICATION 


33 


32.    Remembering  that  the  multiplicand  is  the  same  as 
the  repeated  addend,  answer  the  following : 

1.  Can  you  multiply  16  by  3?    3  by  |6?   6  ft.  by  3 ft.? 

2.  Can  you  find  8  lb.  X  2 ?  2x8?  2  ft.  x  3  ft.  ? 

3.  When  the  multiplicand  is  some   number   of  yards, 
what  is  the  product? 

4.  Can  the  multiplier  ever  be  concrete  ?     Why  ? 

5.  Which  is  more,  |6  x  3  or  |3  x  6? 


33.  Table  of  Products  and  Quotients. 

For  reference  only. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

C 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

"28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

6Q 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

Note.  If  the  pupils  are  not  thoroughly  familiar  with  the  facts 
of  multiplication  and  division,  a  thorough  mastery  of  these  facts 
should  precede  the  attempt  to  use  them  in  the  exercises  that  follow. 
For  a  systematic  method  of  teaching  these  facts,  see  Elementary  Arith- 
metic of  this  series. 

AR.— 3 


84  REVIEW  OF  INTEGERS  AND  DECIMALS 

34.   Oral  Exercises. 

Supply  products  for  x^  and  add  to  each  product  the 
numbers  above  the  column,  as  in  a  (products  12,  16,  35, 
etc.);  adding  2:  14, 18,  37,  etc. ;  adding  6:  18,  22,  41,  etc. 


a 

h 

c 

d 

^x2  =  x 

3,  5 
9x3  =  2; 

4,  7 
7x7  =  2; 

8,  9 
7  x4  =  2; 

4x4  =  2; 

8x5  =  2; 

8x9  =  2; 

7x5  =  2; 

5  X  1  =  x 

4x9  =  2; 

6x6  =  2; 

4x5  =  2; 

8x4  =  2: 

6x7  =  2; 

7x9  =  2; 

9x8  =  2; 

8x7  =  2; 

9x5  =  2; 

6  x4  =  2; 

5  x5  =  2; 

9  x4  =  2; 

6  x8  =  2; 

9x6  =  2; 

4x8  =  2; 

5x6  =  2; 

9x9  =  2; 

8x8  =  2; 

3x9  =  2; 

9x3  =  2; 

7  X  8  =  2; 

8x7  =  a; 

8x6  =  a; 

35.  Written  Exercises. 
1.    Multiply  25,435  by  304. 
Model:     25435         Explanation:    25435 

304  304 

101740  101740  =     4  times  25435 

76305  7630500  =  300  times  25435 

7732240  7732240  =  304  times  25435 

Multipl}^  by  4;  multiply  by  3  (that  is,  300),  placing  the  product 
of  3  and  5  directly  below  the  3,  or  in  hundreds'  place.     Add. 

Solve : 

2.  435,450x504  7.  $386.50x527* 

3.  978,689x450  8.  $868.75x689 

4.  230,302  X  800  9.  $768.30  x  843 

5.  967,843  X  769  10.  $364.97  x  107 

6.  845,397x896  11.  $426.87x489 

♦  Point  off  two  places  for  cents  in  the  answer. 


MULTIPLICATION  35 

36.  Oral  Exercises.* 

1.  How  much  will  4  sheep  cost  at  $5  each? 

Model  for  oral  recitation  :  Since  1  sheep  costs  $5,  4  sheep  will  cost 
4  times  $5,  or  120. 

2.  How  much  will  4  chairs  cost  at  $3  each  ? 

3.  At  f 2  each,  how  much  will  6  books  cost? 

4.  There  are  4  quarts  in  a  gallon.  How  many  quarts 
are  there  in  5  gallons  ? 

5.  Make  and  solve  ten  similar  problems  in  multiplication. 

37.  Written  Exercises. 

1.  Find  the  cost  of  32  acres  of  land  at  $75  an  acre. 

Model  ;    Statement :  1 75  X  32  =  a; 

Work:  $75,  cost  of  1  acre. 
32 
150 
225 
$2400,  cost  of  32  acres. 

2.  If  a  boy  attends  school  180  days  each  year  for  12 
years,  how  many  days  will  he  attend  in  12  years  ? 

3.  Find  the  cost  of  25  cows  at  $45  each. 

4.  If  a  man  earns  $1.75  a  day,  how  much  will  he  earn 
in  24  days  ? 

5.  Make  and  solve  ten  problems  in  multiplication  simi- 
lar to  those  a  clerk  in  a  grocery  store  has  to  solve. 

*  Drill  should  be  given  upon  these  and  similar  problems  until  the  pupils 
are  familiar  with  the  language  forms  used  in  the  analysis.  The  written 
form  should  be  taken  up  after  the  oral  form  has  been  mastered.  Apply 
this  form  of  analysis  (or  some  suitable  form)  to  similar  problems  on  the 
succeeding  pages  of  the  text.  When  tlie  form  has  been  mastered,  the 
pupils  should  be  permitted  to  reply  briefly,  thus  for  Prob.  1  :  Four  times 
five  dollars,  or  twenty  dollars. 


86  REVIEW   OF  INTEGERS  AND  DECIMALS 

REVIEW  — FARM   PROBLEMS 

38.  1.  A  man  bought  a  farm  of  160  acres  at  i  75  an 
acre.     Find  the  cost  of  the  farm. 

2.  During  the  first  year  he  expended  the  following 
sums  for  improvements  :  repairing  fences,  $165.80;  dig- 
ging a  well,  $95;  building  a  carriage  house,  $640;  re- 
shingling  the  barn,  $124.35;  draining  a  marsh,  $60. 
Find  the  cost  of  the  improvements. 

3.  The  farm  was  divided  into  5  fields  of  20  acres 
each,  3  fields  of  10  acres  each,  10  acres  of  orchard,  5  acres 
for  yards  and  garden,  and  the  rest  was  timber  land.  How 
many  acres  of  timber  were  on  the  farm  ? 

4.  To  stock  up  the  farm,  the  farmer  bought  14  head  of 
cattle  at  an  average  of  $  37  apiece,  5  horses  at  an  average 
of  $120  apiece,  24  sheep  at  $4.50  apiece,  6  hogs  at  $5.25 
apiece,  and  30  chickens  at  $.35  apiece.    Find  the  cost  of  all. 

5.  The  following  amounts  were  received  from  the  sale 
of  milk  for  one  year:  Jan.,  $60.80;  Feb.,  $68.17;  March, 
$70.30;  April,  $71.90;  May,  $79.25;  June,  $80;  July, 
$72.35;  Aug.,  $66.10;  Sept.,  $63.28;  Oct.,  $59.37;  Nov., 
$50.40;  Dec,  $54.30.  Find  the  amount  received  during 
the  year. 

6.  The  farmer  employed  one  man  for  8  months,  paying 
him  $35  a  month,  and  another  man  for  3  months,  paying 
him  $  38  a  month.     Find  the  amount  expended  in  wages. 

7.  Two  of  the  20-acre  fields  were  sown  in  oats,  and 
the  yield  was  45  bu.  to  the  acre.  If  oats  were  worth  34^ 
per  bushel,  find  the  value  of  the  crop. 

8.  During  the  month  of  April  the  farmer  sold  $15.75 
worth  of  eggs.  At  the  same  rate,  how  much  would  the  sale 
of  eggs  amount  to  in  one  year? 


MULTIPLICATION  37 

39.  Multiplication  and  Division  by  10,  100,  etc. 

1.  Compare  the  value  of  2  in  20  and  in  2;  in  200  and  in 
20 ;  in  2000  and  in  200. 

2.  What  effect  upon  the  value  of  a  figure  has  (a)  mov- 
ing it  one  place  to  the  left  ?  (J)  moving  it  two  places  to 
the  left  ?    (c)  moving  it  three  places  to  the  left  ? 

3.  Annexing  a  cipher  to  an  integer  has  the  effect  of 
moving  the  digits  each  one  place  to  the  left.  This  mul- 
tiplies the  number  by  10.  Annexing  two  ciphers  has 
what  effect  upon  the  places  occupied  by  the  digits  ?  State 
a  short  method  of  multiplying  an  integer  by  100 ;  by  1000. 

4.  Using  the  short  method,  multiply  each  by  10;  by 
100;  by  1000:  6,  47,  390,  20,  475,  8,  600,  72,  25,  64,  640. 

5.  State  how  an  integer  may  be  multiplied  by  10;  by 
100;  by  1000. 

6.  Moving  a  figure  one  place  to  the  right  has  what 
effect  upon  its  value  ?  Compare  the  value  of  6  in  60  and 
in  6;  in  600  and  in  6;  in  6000  and  in  6. 

7.  Dropping  the  cipher  at  the  right  of  60  changes  the 
number  to  6.  What  change  does  this  make  in  the  value 
of  the  number  ?  State  a  short  method  of  dividing  an  in- 
teger ending  in  a  cipher  by  10. 

8.  What  change  is  made  in  the  value  of  400  by  drop- 
ping the  two  ciphers  ?  State  a  short  method  of  dividing 
a  number  ending  in  two  ciphers  by  100. 

9.  Divide  each  by  10 ;  by  100:   4500,700,400,  3700. 

10.  An  integer  that  does  not  end  in  a  cipher  may  be 
divided  by  10  by  placing  a  decimal  point  to  the  left  of 
the  right-hand  figure,  thus:  87  divided  by  10  is  8.7.  Is 
this  the  same  as  moving  each  digit  one  place  to  the  right  ? 


88  REVIEW  OF  INTEGERS  AND  DECIMALS 

11.  An  integer  that  does  not  end  in  two  ciphers  may  be 
divided  by  100  by  placing  a  decimal  point  at  the  left  of 
the  figure  in  tens'  place,  thus:  475  divided  by  100  is  4.75. 

12.  Give  the  quotient  of  each  divided  by  10;  by  100: 
325,  560,  4582,  4500,  48,  4,  2,  10,  5,  50. 

13.  Write  integers  and  divide  each  by  10  ;  by  100. 

*       40,    Short  Methods.    (  To  be  used  in  subsequent  work. ) 

1.  What  part  of  100  is  25  ?  Compare  25  times  a  num- 
ber with  100  times  the  number. 

2.  To  multiply  by  25,  multiply  by  100  and  divide  by  4. 
Multiply  $489  by  25. 

Model  :       $  489  Explanation  :  Write  as  in  multiplication. 

25  Mentally  multiply  $489  by  100.     Divide  the 

$  12225  product  by  4. 

3.  Multiply  by  25:  1680,11225,  5280  ft.,  231  mi., 

187.56,1247.82. 

4.  Multiply  7865  by  369. 

Model:        7865 

Explanation  :  First  multiply  by  9.    As 

70785  36  is  4  times  i),  multiply  70786  by  4,  writ- 

283140  ^"S  ^^®  product  as  in  the  model.     Add. 

2902185 

5.  In  multiplying  by  84,  first  multiply  by  4  ;  then  mul- 
tiply this  product  by  2,  writing  the  first  figure  of  the  prod- 
uct in  tens'  place.  State  how  you  would  multiply  by  63; 
by  126  ;  by  246  ;  by  729;  by  279.     Illustrate  each. 

6.  Multiply  6840  by  248 ;  by  328 ;  by  648 ;  by  168. 

7.  Multiply  1840  by  287 ;  by  147 ;  by  637 ;  by  639. 


MULTIPLICATION  39 

4:1.  Multiplication  of  Decimals. 

1.  Find  the  sum  of  5.2  mi.,  5.2  mi.,  and  5.2  mic  This 
sum  is  the  same  as  the  product  of  5.2  mi.  x  3.  How 
many  decimal  places  are  there  in  this  product?     Why? 

2.  Find  the  sum  of  5.2  mi.,  5.2  mi.,  5.2  mi.,  5.2  mi.,  and 
5.2  mi.  Multiply  5.2  mi.  by  5.  How  many  decimal  places 
are  there  in  the  product?     Why? 

3.  Write  6.08  x  3  in  the  form  of  addition  and  find  the 
sum.  Multiply  6.08  by  3.  How  many  decimal  places  are 
there  in  the  product?     Why? 

4.  Write  6.08  X  5  in  the  form  of  addition  and  find  the 
sum.  Multiply  6.08  by  5.  Compare  the  results.  How 
many  decimal  places  are  there  in  the  result?     Why? 

5.  State  a  short  method  of  multiplying  an  integer  by  10. 
6.25  may  be  multiplied  by  10  by  moving  the  decimal  point 
one  place  to  the  right.  6.25  x  10  is  62.5.  Has  moving 
the  decimal  point  one  place  to  the  right  the  same  effect  as 
moving  the  digits  one  place  to  the  left?  Compare  2.2 
with  22.     Compare  75.25  with  752.5. 

6.  State  a  short  method  of  multiplying  a  decimal  by 
100.  How  many  places  to  the  right  must  the  decimal 
point  be  moved  to  multiply  by  10?    by  100?  by  1000? 

7.  Compare  25  with  2.5.  Here  the  digits  have  been 
moved  one  place  to  the  right.  Compare  2.5  with  .25. 
Here  the  digits  have  been  moved  another  place  to  the 
right.  This  has  been  done  by  moving  the  decimal  point 
one  place  to  the  left.  Moving  the  decimal  point  one 
place  to  the  left  has  what  effect  upon  the  value  of  a 
decimal?  upon  the  value  of  an  integer? 

8.  State  a  quick  way  of  dividing  a  decimal  by  10  ; 
by  100.     Illustrate  with  integers  and  with  decimals. 


40  REVIEW  OF  INTEGERS  AND  DECIMALS 

42.   Oral  Exercises. 

1.    Divide  each  by  10  and  by  100:  450,  25.74,  45,  .4, 
4.5,  346.2. 
0  2.    What  is  6  times  4?  1  times  4?  J  of  4  ?  ^  of  4  ? 

3.  What  is  meant  by  4x6?  4x1?  4xJ?  4x.l? 

4.  4  X  .1  is  the  same  as  4  divided  by  v^^hat  number? 
4  X. 1  =  37.     4x.2  =  a:.     4x.3  =  a;. 

5.  Divide  4  by  100.  What  is  meant  by  4  x  .01  ?  4  x  .01 
=  x.     4  X  ,06  =  x. 

6.  What  is  meant  by  .4  x  2?  .4  x  13?  .4  x  .1?  .4  x  .1 
is  the  same  as  .4  divided  by  10.  .4  -s- 10  =  a;.  .4  x  .1  =  a;. 
.4  X  .2  —  x.     .4  X  .5  =  x, 

7.  Multiply  82.30  by  10.  Multiply  12.36  by  100. 
Multiply  12.30  by  .1.  Multiply  $24.50  by  .01;  by  .1. 
Multiply  145.75  by  10  ;  by  .1 ;  by  100  ;  by  .01. 

\J  8.  To  multiply  by  .1  is  the  same  as  to  divide  by  10. 
What  change  made  in  the  place  of  the  decimal  point  in 
the  multiplicand  divides  it  by  10  ? 


n 


43.    Multiply  each  by  10  ;  by  .1  ;  by  100  ;  by  .01. 


1. 

$37.50 

5. 

625  ft. 

9. 

2240  lb. 

13. 

3.1416 

2. 

$2500 

6. 

1726  yd. 

10. 

2000  lb. 

14. 

24  cwt. 

3. 

14.525 

7. 

5280  ft. 

11. 

625  lb. 

15. 

4.75  cvvt. 

4. 

17500 

8. 

2150.42 

12. 

630  lb. 

16. 

20  T. 

^    17.    624  X  .001  =  a:.     2000  lb.  x  .001  =  a:. 

18.  Divide   by   100:    4632  lb.;    3000  lb. ;  2160  mi.; 
37.40  mi.;  $234.50;  .425  mi.;   .03  mi.;    3.1416  ft.;   $60. 

19.  At  $5  per   hundredweight,    how    much    is    sugar 
worth  per  pound  ?  at  $4.75  per  cwt.  ?  at  $4.50  per  cwt.  ? 

20.  Multiply  each  in  the  shortest  way  :  5280  ft.  X  25  ; 
1728  X  25  ;  987,647  by  648  ;  7,389,675  by  369. 


MULTIPLICATION  41 

44,  Written  Exercises. 

1.    Multiply  6.23  by  4.2. 

Model  :    Explanation  : 

6.23  6.23  First  multiply  6.23  by  .2.     This  is  equiva- 

.  ^  .  ^  l^iit  to  dividing  6.23  by  10  and  multiplying 

-^  ^  ^  the  quotient  by  2.     6.23  --  10  =  .628 ;   .623 

1246  1.246  ^2  =  1.246.      Next,   multiply  6.23    by  4. 

2492  24.92  6.23  x  4  =  24.92.     Add  the  products. 

26.166  26.166 

Notice  that  in  the  above  the  number  of  decimal  places 
in  the  product  is  the  same  as  the  sum  of  the  number  of 
decimal  places  in  the  multiplicand  and  multiplier.  As 
this  is  always  true,  the  following  method  may  be  employed  : 

To  multiple/  decimals^  multiple/  as  in  integers  and  point 
off  in  the  product  as  many  decimal  places  as  there  are  in 
both  multiplicand  and  multiplier. 

Solve.     Estima^  each  result  before  multiplying: 


2. 

59.786 

X8.97 

r- 

.0056  X  385.07 

3. 

487.69 
53.008 

x.479 
X  7.086 

\- 

7.0758  X  67.09 

4. 

9. 

.07854  X  8.0065 

5. 

.69387 

X  6.9075 

10. 

46,897x4.008 

6. 

13.006 

X  3.1416 

11. 

785.06  X  6300 

45.  Written  Exercises. 

1.  If  a  train  travels  at  an  average  rate  of  43.5  mi. 
an  hour,  how  far  will  it  travel  in  24  hours  ? 

2.  The  circumference  of  a  circle  is  3.1416  times  its 
diameter.  Find  the  circumference  of  a  circle  that  is  9.5 
in.  in  diameter. 

3.  If  it  costs  a  boy  $.50  a  week  to  keep  a  pony,  how 
much  will  it  cost  to  keep  it  for  1  year  (52  wk.)? 


42 


REVIEW   OF  INTEGERS  AND  DECIMALS 


BILLS  AND  ACCOUNTS 

46.  A  Receipted  Bill. 

Los  Angeles,  Cal.,  May  31,  1907. 

Mr.  James  J.  Davies,  2217  Vine  St. 

In  account  with  S.  D.  James  &  Co. 


May 

3 

Z  lb.  coffee            $.40 
2  lb.  tea                   M 
6  bars  soap               .05 

$1 
1 

20 
30 
30 

(4 

5 

3  cans  corn              .08 
1  doz.  lemons          .15 

24 
15 

(( 

6 

10  lb.  sugar            .06 
Received  Payment^ 

60 

$3 

79 

S.  D.  James  ^  Co. 

1.  The  person  who  buys  on  account  is  called  the  debtor, 
and  the  person  who  sells  on  account  is  called  the  creditor. 

2.  Each  purchase,  or  payment,  is  an  item.  How  many 
items  of  debit  are  there  in  this  bill  ?    of  credit  ? 

3.  A  bill  must  show  the  date  of  each  transaction.  "When 
was  the  above  bill  made  out,  or  rendered? 

4.  A  bill  must  also  name  the  debtor  and  the  creditor 
and  the  several  items  of  debit  and  credit.  Who  is  the 
debtor  named  in  the  above  bill  ?     Who  is  the  creditor  ? 

5.  When  a  bill  is  paid,  the  creditor  writes  "  Paid  "  or 
"  Received  payment  "  on  the  bill  and  signs  his  name  below. 
Has  the  above  bill  been  paid? 

6.  Should  a  person  make  a  practice  of  keeping  receipted 
bills  ?     Why  ? 


MULTIPLICATION  43 

47.   Written  Exercises. 

Make  out  and  receipt  the  following  bills.  Supply  all 
necessary  data  not  contained  in  the  problems: 

1.  Mr.  J.  S.  White,  residing  at  234  First  Street,  bought 
of  the  grocery  firm  of  Allen  and  Baker  the  following  : 
May  25,  1907,  2  lb.  tea  @  60^;  3  lb.  coffee  @  45/;  2  lb. 
bacon  @  20 /;  1  lb.  butter  @  30 /.  On  May  31,  4  cans 
tomatoes  @  8/;  2  doz.  eggs  @  18j^;  1  lb.  cheese  @  20  j^. 

2.  Mrs.  Harry  Smith,  residing  at  1450  Jackson  Street, 
bought  of  Cole  Bros,  the  following :  April  24,  1907,  9  yd. 
silk  @  $1.50  ;  2  yd.  dress  lining  @  25^;  4  spools  silk  @ 
10/;  1  bolt  skirt  binding,  15/;  dress  trimmings,  $1.50. 
The  bill  was  rendered  April  30. 

3.  Insert  your  own  name  as  purchaser  of  the  following 
bill  of  hardware:  1  garden  rake,  45/;  1  shovel,  60 /;  3 
lb.  nails  @  6/;  12  yd.  wire  netting  @  30 /;  2  lb.  staples 
@  5/;  1  lawn  mower,  12.50. 

4.  Dr.  C.  L.  Ward  employed  a  schoolboy  to  take  care 
of  his  horses,  for  which  he  agreed  to  pay  him  $6  per 
month,  with  extra  pay  for  additional  services.  During 
the  month  of  May  the  boy  mowed  the  lawn  twice,  for 
which  he  was  to  receive  50  /  each  time  ;  and  he  also 
worked  12  hr.  in  the  garden,  at  10  /.  per  hour.  At  the 
end  of  the  month  Dr.  Ward  asked  the  boy  to  render 
his  bill  for  services  during  the  month.  Make  out  the 
above  bill,  using  your  own  name  or  the  name  of  some  boy 
in  your  school  as  the  creditor.    - 

5.  Make  out  a  bill  for  8  music  lessons  at  $1.50  each. 

6.  Make  out  a  bill  for  purchases  at  a  furniture  store. 

7.  Make  out  a  bill  for  purchases  at  a  meat  market. 

8.  Make  out  a  bill  for  purchases  at  a  dry  goods  store. 


k 


44  REVIEW  OF  INTEGERS  AND  DECIMALS 

RECEIPTS 

48.  1.    Explain  the  meaning  of  the  following : 

Oakland,  Cal.,  May  1, 1907. 

Received  of  Mr.  C.  W.  Smith  twenty-five  dollars 
($  25)  in  full  for  rent  of  house  at  704  Logan  Avenue  for 
the  month  of  May,  1907.  D.  S.  Stone. 

2.  E.  M.  Day  bought  a  sewing  machine  of  P.  Orr 
for  |;30.     Write  a  receipt  for  the  payment  of  this  amount. 

3.  Mr.  J.  E.  Thomas  rented  a  farm  of  D.  R.  James  for 
$  300  per  year.  Write  out  a  receipt  for  the  payment  of 
rent  for  the  year  beginning  March  1,  1905. 

4.  The  manager  of  an  athletic  club  received  $7.50  from 
the  treasurer  of  the  club.     Write  the  receipt. 

49.  Oral  Exercises. 

Whenever  the  partial  sum  is  10,  20,  etc.,  take  the  next 
two  numbers  together,  as  in  <2,  10, 19,  26,  30,  39,  etc. 


a 

b 

c. 

d 

e 

/ 

9 

h 

i 

J 

A; 

Z 

6 

8 

4 

6 

8 

7 

7 

9 

5 

4 

7 

8 

2 

3 

6 

7 

8 

6 

5 

9 

8 

9 

8 

5 

3 

3 

3 

9 

5 

7 

8 

5 

4 

7 

9 

4 

8 

7 

9' 

8 

9 

6 

5 

7 

3 

8 

7 

3 

4 

7 

6' 

3 

8 

7 

7 

9 

8 

5 

5 

8 

5 

3 

6 

8 

8 

4 

8 

9 

5 

7 

8 

9 

4 

3 

8' 

9 

4 

3 

5 

2 

7 

6 

6 

8 

7 

4 

6 

3 

8 

6 

7 

9 

5 

7 

2 

3 

4 

3 

6 

8 

8 

7 

5 

9 

8 

8 

2 

8 

5 

3 

4 

9 

6 

9 

8 

7 

4 

6 

9 

6 

4 

2 

7' 

7 

7 

6 

8 

8 

9 

8 

4 

7 

6 

8 

9 

3 

7 

5 

9 

5 

7 

8 

7 

\) 

^ 


DIVISION  45 

DIVISION  OF  INTEGERS  AND  DECIMALS  ^ 

50.  Factors  and  Multiples. 

1.  4  times  3  are  12.  4  and  8  are  each  a  factor  of  12. 
Name  two  other  factors  of  12;  two  factors  of  14;  of  20. 

2.  2  and  8  are  each  a  factor  of .     3  is  a  factor  of 

.     5  and are  each  a  factor  of  10. 

3.  The  factors  of  a  number  are  the  integers  which,  when 
multiplied,  make  the  number.  A  number  may  have  sev- 
eral pairs  of  factors,  thus  :  the  pairs  of  factors  of  24  are  4 
and  6,  3  and  8,  2  and  12.  Some  numbers  have  only  a 
single  pair  of  factors,  thus  :    the  factors  of  15  are  8  and  5. 

4.  Name  all  the  pairs  of  factors  of  each  of  the  follow- 
ing :  4,  6,  8,  10,  12,  14,  15,  16,  18,  20,  21,  24,  27,  28,  30. 

5.  As  3  times  5  are  15, 15  is  a  multiple  of  both  3  and 
5.     Name  a  multiple  of  both  4  and  5;  of  both  4  and  7. 

6.  The  number  obtained  by  multiplying  together  two 
integers  is  called  a  multiple  of  either  integer.  Name  a 
multiple  of  4;  of  both  6  and  5. 

7.  In  multiplication,  two  factors  are  given  to  find  their 
product.  In  division,  the  product  and  one  of  two  factors 
are  given  to  find  the  other  factor. 

61.  1.  The  process  of  finding  the  other  factor,  when 
the  product  and  one  of  two  factors  are  given,  is  called 
division. 

2.  Since  three  5's  are  15,  the  number  of  5's  in  15  is  3. 

3 
This  may  be  indicated  thus :  5)15     Here  5  is  the  given 
factor,  15  is  the  given  product,  and  3  is  the  other  factor. 

3.  The  given  factor  is  called  the  divisor,  the  given 
product  is  called  the  dividend,  and  the  factor  found  is 
called  the  quotient. 


46  REVIEW   OF  INTEGERS  AND  DECIMALS 

62.  1.  Name  all  the  multiples  of  3  to  30  ;  of  4  to  40 ; 
of  5  to  50;  of  6  to  60;  of  7  to  70;  ofStoSO;  of9to90. 

2.  In  each  of  the  following,  name  the  higliest  multiple : 
a,  Of  2  in  17,  11,  13,  9,  5,  7, 15, 19,  3, 10, 14,  21, 17, 19. 
h.    Of  3  in  10,  8,  14,  19,  57,  16,  20,  29,  23,  26,  17,  22. 

c.  Of  4  in  10,  17,  7,  15,  38,  31,  22,  27,  35,  19,  25,  30. 

d.  Of  5  in  17,  23,  7,  12,  28,  34,  42,  39,  48,  27,  33,  19. 

e.  Of  6  in  16,  10,  29,  22,  38,  45,  51,  34,  57,  43,  53,  41. 
/.  Of  7  in  52,  46,  17,  33,  25,  68,  57,  39,  13,  30,  44,  53. 
ff.  Of  8  in  52,  15,  28,  20,  35,  46,  69,  75,  58,  30,  60,  44. 
7i.  Of  9  in  20,  78,  34,  42,  47,  37,  59,  68,  11,  29,  53,  16. 

3.  Repeat  Ex.  2,  naming  the  highest  multiple  and  the 
difference  between  it  and  the  given  number,  thus  for  g : 
48  and  4 ;  8  and  7  ;  24  and  4,  etc. 

4.  Repeat  Ex.  2,  giving  the  quotients  and  the  remain- 
ders, thus  for  g :  6  and  4  over ;  1  and  7  over ;  etc. 

The  sign  of  division  is  -^.  It  indicates  that  the  num- 
ber before  it  is  to  be  divided  by  the  number  after  it. 
6  ^  3  is  read  6  divided  hy  3.  Division  may  also  be  indi- 
cated thus  :  3)^;  or  thus  :   |. 

53.  Oral  Exercises. 

Supply  the  value  of  x  in  each  of  the  following : 
a  h  c  d 

24-^6  =  a:  58^8  =  a;  67^9  =  a;  76-f-8  =  a? 

30-j-8  =  a:  76^9  =  2:  69-^8  =  a;  61^-9  =  a; 

42-^9  =  ic  40-^7  =  2;  60-j-7=.r  45-!- 6  =  a; 

18-r-7=a;  23^9  =  a:  52  -  6  =  a:  39^  7  =  a: 

25-5-7=2;  70-8  =  2:  43-^9  =  2;  26-^3  =  2: 

36-*- 8  =  2;  52-7  =  2:  59-r-6  =  2?  22-6  =  2: 

51-1-6  =  2;  28^6  =  2:  39^-5  =  2:  50-5-8  =  2: 


DIVISION 

47 

54,  Written  Exercises. 

Use   as    successive  divisors 

the    numbers    above    the 

columns.     Solve 

by  short  division. 

a 

h 

c 

d* 

6,T,8 

9,5,4 

3,8,7 

2,9,6 

1.    672,458 

327,459 

836,594            ^ 

$4387.24 

2.   237,400 

574,063 

683,127 

$3058.26 

3.    946,305 

508,342 

427,060           i 

$9576.30 

4.    375,268 

970,654 

738,967          i  $4256.75 

5.    834,008 

624,307 

520,380 

$5687.50 

6.    463,925 

207,193 

946,425 

$7495.38 

7.    927,384 

423,075 

315,017 

$8250.15 

55.  Written  Exercises. 

Use  as  multipliers  the  numbers  above  the  columns : 


a 

789 

6 
465 

c 
305 

d 
890 

1. 

$8796.50 

7568.93 

975.864 

432.501 

2. 

$4578.69 

835.769 

586.097 

34.2056 

3. 

$9760.50 

58.7964 

4376.89 

Q^Q4.M 

4. 

$6389.75 

95,687.3 

605.008 

5046.32 

5. 

$4975.86 

2456.78 

789.645 

265.423 

6. 

$7869.45 

697.583 

83.7956 

123.456 

7. 

$3698.70 

4309.58 

3945.78 

45.0635 

8. 

$8970.56 

37.6895 

687.905 

5643.06 

9. 

$6875.09 

709.608 

'201.003 

326.504 

.0. 

$3204.56 

854.076 

4567.98 

Q^MbQ 

*  Place  a  decimal  point  in  the  answer  above  the  decimal  point  in  the 
dividend. 


48  REVIEW  OF   INTEGERS  AND  DECIMALS 

56.  Measurement  and  Partition. 

1.  All  division  is  either  measurement  or  partition. 

2.  6  ft.  6  ft. 

_4 6  ft. 

24  ft.  when  expressed  as  addition  is     6  ft. 

6  ft. 


24  ft. 

3.  The  quantity  24  ft.  may  be  measured  by  the 
quantity  6  ft.     The  measure  6  ft.  is  contained  in  24  ft. 

4  times.    This  may  be  indicated  thus  :    ^  „   .r—- 

^  6  ft.)24  It. 

4.  The  process  of  finding  how  many  times  one  number 
or  quantity  is  contained  in  another  is  called  division  by 
measurement. 

5.  In  Ex.  2  the  quantity  6  ft.  is  taken  4  times  to  give 
the  quantity  24  ft.     Therefore,  one  fourth  of  24  ft.  is  6  ft. 

(K  f '■ 

This  may  be  expressed  thus  :   A^^^oTTr 

6.  The  process  of  finding  one  of  the  equal  parts  of  a 
number  or  quantity  is  called  division  by  partition. 

7.  In  every  problem  in  division  you  will  be  required 
to  find  either  (a)  how  many  times  some  unit  of  measure 
is  contained  in  a  quantity  to  be  measured,  or  (5)  to  find 
a  given  part  of  some  quantity  to  be  divided. 

67.  Measurement. 

1.  The  number  of  4  cubes  in  12  cubes  may  be  found 
by  using  the  unit  4  cubes  as  a  measure  to  measure  the 
quantity  12  cubes.  4  cubes,  the  unit  of  measure,  is  con- 
tained in  12  cubes,  the  quantity  to  be  measured,  exactly 

3 
3  times.     This  may  be  indicated  thus:     4  cubes)  12  cubes 


DIVISION  49 

2.  Show  with   objects    or   by  a  diagram   how  many 

3  apples  there  are  in  12  apples.     Indicate  this  in  the  form 
of  division.     Write  the  quotient. 

3.  Show  by  a  diagram  or  by  actual  measurement  how 
many  2  ft.  there  are  in  12  ft.  Indicate  this  in  the  form  of 
division.     Write  the  quotient. 

4.  Show  with  objects  or  by  a  diagram  what  is  meant 
by    each:    2    books)10    books;      5    pencils)15    pencils; 

4  boys)  12  boys;  6  books)  12  books;  3  ft.)9lt7;  5^)20/. 
Write  a  problem  for  each. 

5.  In  finding  the  number  of  4  hr.  there  are  in  24  hr., 
what  is  the  unit  of  measure  ?  What  is  the  quantity  to  be 
measured? 

6.  Show  by  using  books  (a)  how  many  3  books  there 
are  in  9  books ;  (6)  how  many  3  books  there  are  in  10 
books;  (<?)  how  many  8  books  there  are  in  11  books. 
Indicate  each  in  the  form  of  division,  with  quotient,  and 
remainder,  if  any. 

7.  Show  with  objects  that  the  number  of  4  apples  in 
10  apples  is  2,  with  2  apples  remaining. 

8.  Find  the  number  of  $4  there  are  in  |16 ;  of  3  yd. 
there  are  in  15  yd. ;  of  7  da.  there  are  in  21  da.  Indicate 
each  in  the  form  of  division. 

9.  In  division  by  measurement  the  divisor  is  always 
like  the  dividend.  The  quotient  is  always  an  abstract 
number,  since  it  tells  how  many  times  the  unit  of  measure 
is  contained  in  the  quantity  measured. 

10.  Find  by  measuring  how  many  times  a  1-ft.  meas- 
ure must  be  applied  to  measure  4  ft.  (4  ft.  ^  1  ft.  =  a;)  ; 
a  J-ft.  measure  to  measure  4  ft.  (4  ft.  -r- ^  it.  =  x)  ;  a.  J-ft 
measure  to  measure  4  ft.   (4  ft. -t- J  ft.  =  a;). 


60  REVIEW  OF   INTEGERS   AND  DECIMALS 

58.  Oral  Exercises — Measurement. 

1.  At  $  4  each,  how  many  desks  can  be  bought  for  fi  12  ? 
The  unit  of  measure  is  f  4  and  the  quantity  to  be  meas- 
ured is  112. 

Model  for  oral  recitation  :  Since  1  desk  costs  $4,  as  many  desks 
can  be  bought  for  $  12  as  there  are  $  4  in  ^  12,  or  3. 

In  each  of  the  following,  name  the  unit  of  measure  and 
the  quantity  to  be  measured: 

2.  At  f  3  each,  how  many  chairs  can  be  bought  for  $  15  ? 

3.  At  1 4  a  pair,  how  many  pairs  of  shoes  can  be  bought 
for  1 24? 

4.  If  berries  cost  6  ^  a  box,  how  many  boxes  of  berries 
can  be  bought  for  30  ^  ? 

5.  How  many  yards  of  ribbon  at  8  ^  a  yard  can  be 
bought  for  40^? 

6.  If  a  boy  earns  1 9  a  month,  in  how  many  months  will 
he  earn  f  45  ? 

7.  Write  ten  additional  problems  in  measurement. 

69.  Written  Exercises  —  Measurement 

1.  At  $  9  a  ton,  how  many  tons  of  hay  can  be  bought 
for  1 216? 

2.  A  man  bought  sheep  at  $6  each.  He  paid  $96  for 
all.     How  many  sheep  did  he  buy  ? 

3.  A  farmer  divided  a  farm  containing  160  acres  into 
10-acre  fields.  Find  the  number  of  fields  and  the  value  of 
each  field  at  $  80  per  acre. 

4.  How  many  9's  are  there  in  1728  ?  Is  this  measure- 
ment? 

5.  Write  five  problems  in  division  by  measurement,  and 
solve  each. 


DIVISION  51 

60.  Partition. 

1.    One  fourth  of  12  cubes  may  be  found  by  dividing 
12  cubes  into  4  equal  groups  or  parts. 

One  fourth  of  12  cubes  is  3  cubes.     This  may  be  in- 
dicated thus : 

3  cubes 


4)12  cubes 

2.  Show  with  objects  and  by  diagrams  what  is  meant 
by  one  third  of  12  objects  ;  by  one  fourth  of  8  objects  ;  by 
one  third  of  9  objects.  Indicate  these  in  the  form  of 
division,  and  write  the  quotients. 

3.  Show  with  objects  and  by  a  diagram  what  is  meant 
by  each  of  the  following  : 


4)8  apples      2)10  in.      5)10  in.      3)15  ft.      4)16  books. 

4.  In  division  by  partition  the  quotient  is  always  a 
part  of  the  dividend.  Therefore,  the  quotient  is  always 
like  the  dividend,  —  concrete  when  the  dividend  is  con- 
crete, and  abstract  when  the  dividend  is  abstract.  The 
divisor  is  always  an  abstract  number.     Why  ? 

61.  Oral  Exercises  —  Partition. 

1.  If  2  chairs  cost  f  8,  what  is  the  cost  of  1  chair  ? 

Model  for  oral  recitation  :  If  2  chairs  cost  $  8,  1  chair  will  cost 
one  half  of  $  8,  or  $  4. 

Name  in  each  the  quantity  to  be  divided  and  the  num- 
ber of  parts  into  which  it  is  to  be  divided : 

2.  If  2  tables  cost  $  12,  what  is  the  cost  of  1  table  ? 

3.  If  3  stoves  cost  115,  what  is  the  cost  of  1  stove  ? 

4.  At  $  12  a  ton,  what  is  the  cost  of  one  half  ton  of  hay? 

5.  Write  ten  additional  problems  in  partition. 


62  REVIEW  OF   INTEGERS   AND   DECIMALS 

62.    1.    Show  by  a  diagram  or  with  objects  the  mean- 
ing of  : 

2)f6  4)12  da.  3)9l5r  i3)f9  .f45fl2 

2.    Write  a  problem  for  each  : 

3)il8  4j20?  2)10  yd. 


$6)112  I7)$21  5)10  yd. 

3.  4)  1 2  may  be  either  partition  or  measurement.  State 
what  is  meant  by  4)12  (a)  when  it  is  partition; 
(5)    when  it  is  measurement. 

4.  Is  the  divisor  ever  concrete  in  partition  ?  Is  the 
quotient  ever  concrete  in  measurement  ?    Give  reasons. 

5.  In  division  by  measurement  the  quotient  is  always 
what  kind  of  a  number  ? 

6.  When  the  divisor  is  a  concrete  number,  is  the  divi- 
sion partition  or  measurement  ? 

7.  Make  ten  problems  in  division,  and  tell  which  are 
partition  and  which  are  measurement. 

8.  Make  problems  for  each  of  the  following.  Tell 
which  are  partition  and  which  are  measurement. 

$2)fl0  25110  4)16  yd. 

8  wk.)16  wk.  3^115  55W 

63.   Oral  Exercises. 

■^3.  indicates  that  12  is  to  be  divided  by  3.     Solve  each  : 

1.  :^  6.   Ap-  11.    ^Q-  16.    ^ 

2.  ■*/  7.     -^8.  12.     ^  17.     -figL 

3.  J^I  8.    ^  13.    ^  18.    ^ 

4.  Y  9.    ^  !*•    \^  19-    ^ 

5.  Y  10.    ^^  15.    5g^  20.    -V- 


DIVISION  53 

64.   Written  Exercises.* 

1.  At  $S  each,  how  many  tables   can  be  bought  for 

8128? 

Model  for  measurement: 

16,  number  bouglit  for  $  128 
cost  of  1  table,  $  8)-|128 

2.  A  man  spent  1 216  in  6  mo.  What  was  the  av- 
erage amount  spent  each  month  ? 

Modp:l  for  partition  :      _^^  .•     i 

^  $86,  spent  m  1  mo. 

6)  $216,  spent  in  6  mo. 

Tell  which  of  the  following  are  partition  and  which  are 
measurement,  and  solve: 

3.  If  a  boy  saves  85  a  month,  in  how  many  months 
will  he  save  1120? 

4.  How  many  tons  of  coal  at  #7  a  ton  can  a  man  buy 
for  8161? 

5.  Five  boys  agreed  to  share  equally  the  expenses  of 
a  camping  trip.  The  trip  cost  them  821.70.  What  was 
each  boy's  share  ? 

6.  A  girl  bought  8  yd.  of  cloth  for  82.56.  How  much 
did  the  cloth  cost  her  per  yard  ? 

7.  A  hardware  merchant  bought  some  stoves  at  89 
each.  His  bill  amounted  to  8198.  How  many  stoves 
did  he  buy  ? 

8.  If  a  boy  worked  65  problems  correctly  in  1  school 
week  (5  days),  what  was  the  average  number  worked 
correctly  each  day  ? 

9  A  dealer  bought  6  copies  of  a  certain  book.  His 
bill  amounted  to  87.50.    What  was  the  price  of  the  book  ? 

*  Give  the  oral  analysis  of  Probs.  1-9. 

7C 


64  REVIEW  OF  INTEGERS  AND  DECIMALS 

65.  Ratio  or  Comparison. 

1.  How  many  times  must  the  measure  2  ft.  be  applied  in 
measuring  6  ft.  ?  The  number  3  expresses  the  ratio,  or 
relation,  of  the  quantity  6  ft.  to  the  unit  2  ft. 

2.  In  measuring  6  ft.  by  2  ft.  the  quantity  to  be  meas- 
ured is  6  ft.,  and  the  unit  of  measure  is  2  ft.  The  ratio  of 
6  ft.  to  2  ft.  is  found  by  dividing  6  by  2. 

3.  What  is  the  ratio  of  8  ft.  to  4  ft.?  of  12  ft.  to  3  ft.? 

4.  What  is  the  ratio  of  6  da.  to  3  da.?  of  24  hr.  to  6 
hr.?   of  25^  to  5^?  of  fl  to  1.25?  of  75^  to  25^? 

5..  What  is  the  ratio  of  100  to  50  ?  to  25  ?  to  10  ? 
to  20  ?  to  5  ? 

6.  In  measuring  2  ft.  by  4  ft.  the  unit  of  measure  is 
4  ft.,  and  the  quantity  to  be  measured  is  2  ft.  The  meas- 
ure 4  ft.  is  applied  J  time  ;  that  is,  one  half  the  measure 
is  applied  in  measuring  2  ft.     The  ratio  of  2  ft.  to  4  ft.  is  ^. 

66.  Draw  on  the  blackboard  lines  the  length  of  the 
quantities  to  be  measured.  Make  measures  the  length 
of  the  units  of  measure  to  be  used  in  measuring  each  line. 
By  applying  the  measure  to  the  line  to  be  measured,  de- 
termine the  ratio  of   the  following: 

1.  Of  2  ft.  to  1  ft.  7.  Of  1^  ft.  to  3  ft. 

2.  Of  2  ft.  to  I  ft.  8.  Of  J  ft.  to  1  ft. 

3.  Of  1  ft.  to  J  ft.  9.  Of  f  ft.  to  2  ft. 

4.  Of  3  ft.  to  J  ft.  10.  Of  IJ  ft.  to  2  ft. 

5.  Of  J  ft.  to  1  ft.  11.  Of  f  ft.  to  J  ft. 

6.  Of  I  ft.  to  2  ft.  12.  Of  I  ft.  to  ^  ft. 

13.  What  part  of  14  da.  are  7  da.?  What  is  the  ratio 
of  7  da.  to  14  da.?  of  14  da.  to  7  da.  ? 


DIVISION  65 

14.  3  in.  is  what  part  of  6  in.?  What  is  the  ratio  of 
3  in.  to  6  in.  ?  The  ratio  tells  the  number  of  times  the 
unit   of  measure  must  be  applied  to  measure  the  given 

quantity.     A  6-in.  measure  must  be  applied times  to 

measure  3  in.    A  3-in.  measure  must  be  applied times 

to   measure  6  in.     The   ratio  of  6   in.  to  3  in.  is . 

The  ratio  of  3  in.  to  6  in.  is . 

15.  What  part  of  the  measure  12  in.  must  be  applied  to 
measure  3  in.?  What  is  the  ratio  of  3  in.  to  12  in.?  of  12 
in.  to  3  in.? 

16.  The  ratio  of  3  yd.  to  some  quantity  is  ^.  What  is 
the  quantity  ? 

17.  The  ratio  of  some  quantity  to  f  2  is  4.  What  is 
the  quantity  ? 

18.  If  4  is  the  ratio  gf  some  amount  to  8  20,  what  is 
the  amount? 

19.  A  J-ft.  measure  was  used  12  times  in  measuring 
the  length  of  a  line.     How  long  was  the  line  ? 

20.  Draw  a  line  of  such  length  that  a  6-in.  measure 
will  be  applied  IJ  times  in  measuring  it.  Prove  your 
work  by  applying  the  measure. 

21.  What  is  the  ratio  of  2  to  8?  of  6  to  2  ?  of  20  to 
5  ?  of  5  to  30  ?  of  8  to  48  ?  of  40  to  8  ? 

22.  Two  tons  of  coal  will  cost  what  part  of  the  cost  of 
8  tons  ?    of  6  tons  ?    of  12  tons  ? 

23.  3  yd.  of  cloth  wdll  cost  what  part  of  the  cost  of  9 
yd.?  of  15  yd.?  of  6  yd.?  of  12  yd.? 

24.  If  the  cost  of  24  yd.  of  cloth  is  given,  how  may  the 
cost  of  8  yd.  be  found  ?  of  6  yd.?   of  4  yd.  ?  of  12  yd.? 

25.  If  the  cost  of  6  sheep  is  $24,  what  is  the  cost  of 
18  sheep?  of  12  sheep?  of  3  sheep?  of  2  sheep? 


^ 


66  REVIEW   OF  INTEGERS  AND  DECIMALS 

67.  Oral  Exercises. 

1.  If  5  desks  cost  120,  how  much  will  7  desks  cost? 
The  quantities  5  desks  and  7  desks  are  measured  by 

the  common  unit  1  desk.  In  solving  this  problem,  first 
find  the  cost  of  the  unit  1  desk.  Next  find  the  cost  of 
the  required  number  of  units. 

Model  for  oral  recitation  :  If  5  desks  cost  $  20,  1  desk  will  cost  ^ 
of  $  20,  or  $4.     Since  1  desk  costs  $  4,  7  desks  will  cost  7  times  $  4,  or 

^28. 

2.  If  4  tons  of  hay  cost  $82,  how  much  will  6  tons  cost  ? 

3.  If  7  tablets  cost  35^,  how  much  will  4  tablets  cost  ? 

4.  At  the  rate  of  5  for  25^,  how  much  will  8  spelling- 
blanks  cost  ? 

5.  A  girl  paid  30^  for  5  yards  of  ribbon.  How  much 
would  8  yards  have  cost  at  the  same  rate  ? 

6.  Make  ten  additional  problems  similar  to  the  above. 

68.  Written  Exercises. 

1.  A  farmer  raised  220  bu.  of  oats  on  4  acres  of  land. 
How  much  at  the  same  rate  would  a  7-acre  field  have 
produced  ? 

2.  If  a  train  travels  138  mi.  in  3  hr.,  at  the  same 
rate,  how  far  will  it  travel  in  8  hr.  ? 

3.  From  a  farm  of  160  acres  8  acres  were  sold  for 
$500.  At  this  rate,  what  was  the  value  of  the  entire 
farm  ? 

4.  7  men  picked  210  boxes  of  prunes  in  1  day.  At  the 
same  rate,  how  many  boxes  would  15  men  liave  picked  ? 

5.  The  expenses  of  a  family  amounted  to  $328.75  for 
5  ino.  At  the  same  rate,  what  would  the  expenses 
amount  to  in  1  yr.  (12  mo.)? 


DIVISION  67 

69.   Oral  Exercises. 

Finding  a  part  of  an  amount,  when  the  amount  is  given : 

1.  Find  I  of  12  ft. 

i2ft 

I I 1 

V     3ft.  3ft  3ft      J      3  ft 

To  find  I  of  12  ft.,  divide  12  ft.  into  4  equal  parts.     Then  |  of  12 
ft.  will  be  3  of  these  parts. 

2.  Show  by  a  diagram  that  ^  of  12  ft.  is  4  ft.  and  that 
f  of  12  ft.  are  8  ft;  that  |  of  8  yd.  are  6  yd. 

3.  Show  with  objects  that  -|  of  10  objects  are  6  ob- 
jects; that  f  of  6  objects  are  4  objects. 

4.  Show  by  a  diagram  that  if  a  board  is  8  ft.  long,  | 
of  the  length  of  the  board  is  6  ft. 

5.  Whatisf  of  $12? 

Model  for  oral  recitation:  Since  \  of  $12  is  $3,  |  of  $12  is  3  times 

$3,  or  $9. 

6.  What  is  f  of  $  15  ?  of  124  ?  of  $30  ?  of  812  ? 

7.  What  is  f  of  10  mi.?  of  25  mi.?  of  35  mi.? 

8.  What  is  I  of  18  lb.?  of  30  lb.?  of  42  lb.?  of  60  lb.? 

9.  At  20^  a  pound,  how  much  will  |  of  a  pound  of 
cheese  cost? 

10.  How  many  months  are  there  in  |  yr.?  in  f  yr.? 

11.  If  Fred  worked  15  problems  and  John  worked  |  as 
many,  how  many  did  John  work  ? 

12.  A  girl  worked  16  problems,  and  |  of  them  were 
correct.     How  many  of  them  were  correct  ? 

13.  How  many  inches  are  there  in  |^  of  a  foot  ? 

14.  At  $8  a  ton,  how  much  will  -I  of  a  ton  of  coal  cost? 

15.  Make  ten  additional  problems  similar  to  the  above. 


58  REVIEW  OF  INTEGERS  AND  DECIMALS 

70.  Written  Problems. 

1.  There  are  2000  pounds  in  a  ton.  How  many  pounds 
are  there  in  |  of  a  ton  of  hay  ? 

2.  There  are  320  rods  in  1  mile.  How  many  rods  are 
there  in  |  of  a  mile  ? 

3.  A  girl  read  a  book  containing  210  pages.  How  many 
pages  had  she  read  when  she  had  read  |  of  the  book  ? 

4.  Two  boys,  Henry  and  Frank,  bought  out  a  news- 
paper route  that  cost  them  §4.50.  Frank  paid  |  of  the 
cost  of  the  route  and  Henry  paid  J  the  cost.  Kow 
much  did  each  pay  ? 

5.  A  man  had  320  acres.  He  rented  |  of  his  land. 
How  many  acres  did  he  rent  ? 

6.  There  are  6280  feet  in  1  mile.  How  many  feet 
are  there  in  |  of  a  mile? 

71.  Dividing  by  20,  30,  40,  200,  300,  etc. 

1.  State  a  short  method  of  dividing  a  number  by  10  ; 
by  100. 

2.  Divide  476  by  40. 
Model  : 

11.9  First  place  a  decimal  point  in  the  quotient  above  and 

40^476         between  7  and  6.     Then  divide  by  4. 

3.  State  how  you  would  divide  a  number  by  50  ;  by 
400  ;  by  4000. 

Solve.     Before  dividing,  estimate  each  quotient : 

4.  324 -H  40  8.    1260 -f- 400        12.    4860 -i-   40 

5.  1728-*- 60  9.    5280 -f- 600        13.    3600  +  900 

6.  1720-!- 80        10.    7854-^-500        14.    4240 -^  800 

7.  320  +  20        11.    6250  +  500        15.    1240+    30 


DIVISION  59 

72.  Oral  Exercises. 

Finding  an  amount,  when  part  of  the  amount  is  given  : 
1.    When  I"  of  the  length  of  a   board  is  8  ft.,  what  is 
^  of  the  length  of  the  board  ? 


8ft 
If  I  of  the  length  of  a  board  is  8  ft.,  J  of  the  length  of 
the  board  is  what  part  of  8  ft.  ?     If  J  of  the  length  of  a 
board  is  4  ft.,  what  is  |  of  the  length  of  the  board  ? 

2.  Show  by  a  diagram  that  if  \  of  the  length  of  a  line 
is  6  ft.,  I  of  the  length  of  the  line  is  2  ft.  If  \  of  the 
length  of  a  line  is  2  ft.,  what  is  the  length  of  the  line  ? 

3.  Show  by  a  diagram  that  if  |  of  a  line  is  6  ft.  long, 
\  of  the  line  is  2  ft.  long  and  the  line  is  8  ft.  long. 

4.  Using  12  objects,  show  that  since  \  of  12  objects  is 
4  objects,  I  of  12  objects  are  8  objects. 

5.  Show  that  since  |  of  12  objects  are  8  objects,  \  of 
12  objects  is  \  of  8  objects. 

6.  Draw  a  diagram  to  show  the  length  of  a  room,  if  | 
of  the  length  of  the  room  is  9  ft. 

7.  If  112  is  I  of  the  cost  of  a  suit  of  clothes,  what  is 
\  of  the  cost  of  the  suit?     What  is  the  cost  of  the  suit? 

Model  for  oral  recitation  :  If  $12  is  |  of  the  cost  of  a  suit,  \  of  the 
cost  of  the  suit  is  \  of  $12,  or  $4.  Since  $4  is  |  of  the  cost  of  a  suit, 
the  cost  of  the  suit  is  4  times  $4,  or  $16. 

8.  If  ^20  is  I  of  the  cost  of  a  cow,  what  is  the  cost 
of  the  cow? 

9.  If  I  of  the  cost  of  a  book  is  40^,  what  is  the  cost  of 
the  book? 


60  REVIEW  OF  INTEGERS  AND  DECIMALS 

10.  Two  boys  together  bought  a  baseball.  One  boy- 
paid  ^.60,  which  was  |  of  the  cost  of  the  ball.  How  much 
did  the  other  boy  pay  ?    What  was  the  cost  of  the  ball  ? 

11.  A  boy  spent  90/,  which  was  |  of  the  whole  amount 
of  money  he  had.  How  much  money  had  he  ?  How  much 
money  had  he  left  ? 

12.  Fred  weighs  100  lb.     This  is  f  of  George's  weight 
How  much  does  George  weigh  ? 

13.  Make  ten  additional  problems  similar  to  the  above. 
73.  Oral  Exercises. 

Two  addends  whose  sum  is  10  or  less  may  be  taken  as 
a  single  addend.  Exercise  a  below  may  be  added :  13,  23, 
30,  47,  54,  62,  70.     Add  h-m  in  a  similar  manner  : 


a 

b 

c 

d 

e 

/ 

S' 

Ti 

1 

J 

fc 

I 

m 

8 

8 

8 

7 

2 

3 

8 

7 

3 

2 

3 

2 

4 

4 

6 

1 

6 

5 

6 

5 

4 

6 

7 

3 

1 

7 

4 

1 

3 

2 

2 

7 

3 

5 

2 

7 

7 

5 

2 

2 

3 

2 

2 

5 

6 

8 

8 

9 

2 

3 

6 

4 

5 

2 

4 

2 

2  ■ 

3 

5 

5 

7 

5 

5 

8 

8 

8 

4 

5 

5 

5 

3 

3 

4 

6 

2 

4 

3 

3 

9 

2 

2 

4 

2 

6 

5 

7 

2 

7 

3 

4 

1 

3 

3 

6 

7 

5 

8 

9 

4 

9 

8 

3 

2 

4 

4 

6 

7 

8 

7 

6 

5 

5 

2 

7 

8 

o 

-j 

6- 

2 

2 

6 

2 

3 

3 

7 

6 

2 

3 

7 

1 

4 

3 

2 

3 

5 

9 

8 

9 

7 

6 

3 

8 

2 

5 

8 

7 

7 

3 

3 

3 

5 

2 

2 

9 

3 

3 

8 

6 

9 

4 

4 

T) 

5 

4 

6 

7 

8 

7 

6 

Write  ten  columns  in  which  two  addends  whose  sum  is 
10  or  less  may  be  taken  as  a  single  addend.  Add  your 
columns. 

Add  the  columns  in  Sec.  13. 


REVIEW  61 

REVIEW 
74.  Multiplication. 

Name  the  multiples  of  2  to  24  ;  of  3  to  36  ;  of  4  to  48  ; 
of  5  to  60 ;  of  6  to  72  ;  of  T  to  84  ;  of  8  to  96 ;  of  9  to  108. 

Multiply  the  numbers  in  each  column  by  the  number  at 
the  head  of  the  column,  and  add  to  each  product  the  num- 
ber in  parentheses : 


a 

h 

c 

d 

e 

/ 

9 

^ 

2(9) 

3(2) 

4(3) 

5(4) 

6(5) 

7(6) 

8(7) 

9(8) 

7 

6 

5 

4 

9 

8 

2 

8 

3 

3 

8 

2 

4 

4 

6 

2 

9 

2 

4 

5 

7 

5 

9 

9 

4 

7 

2 

8 

2 

7 

3 

3 

8 

9 

9 

3 

3 

6 

4 

7 

5 

8 

7 

6 

8 

2 

8 

4 

9 

4 

6 

9 

6 

9 

T 

6 

2 

5 

3 

7 

5 

3 

5 

5 

6 

7 

8 

9 

7 

6 

9 

7 

8 

9 

6 

6 

9 

8 

8 

6 

6 

6 

9 

8 

6 

7 

6 

8 

7 

8 

7 

7 

8 

9 

7 

9 

75.  Write  the  following  in  the  forms  of  bills,  and  find 
the  amounts  in  each.   (See  p.  42.) 

1.  Jan.  2,  1907  :  3  bars  of  soap  at  6^  each  ;  4  lb.  of 
prunes  at  8^  per  pound;  85  lb.  of  potatoes  at  2^  per 
pound.  Jan.  5,  1907  :  2  lb.  of  coffee  at  33^  per  pound; 
2  lb.  of  cheese  at  18^  per  pound  ;  3  lb.  of  tea  at  ^^^  per 
pound. 

2.  Jan.  9,  1907:  7  yd.  dress  cloth  at  $1.20  per  yard  ; 
1  doz.  handkerchiefs  at  11.40  per  dozen  ;  3  shirts  at  $1.75 
each.  Jan.  10,  1907  :  5  pair  socks  at  $.25  each  ;  1  um- 
brella at  $2.40  ;  1  pair  scissors  at  $.75.  i 


62  REVIEW   OF  INTEGERS  AND  DECIMALS 

76.  Division. 

1.  Name  the  highest  multiple  of  tlie  number  at  the  head 
of  the  column  in  each  number  in  the  column,  thus  for 
column  a  :  18,  9,  15,  etc. 

a  h  G  d  e  f  g 

3  4  5  6  7  8  9 


20 

17 

38 

39 

40 

30 

40 

11 

10 

17 

28 

25 

46 

78 

16 

23 

26 

11 

58 

37 

81 

8 

25 

14 

22 

13 

22 

86 

25 

31 

23 

58 

32 

76 

68 

14 

14 

49 

46 

65 

53 

23 

23 

19 

33 

33 

39 

60 

59 

17 

27 

43 

52 

52 

29 

60 

28 

38 

24 

16 

18 

69 

51 

19 

29 

39 

40 

60 

18 

34 

22 

34 

28 

50 

34 

59 

48 

13 

21 

34 

55 

68 

14 

70 

29 

39 

63 

63 

87 

97 

75 

24 

26 

32 

70 

48 

50 

89 

32 

43 

54 

57 

29 

17 

110 

2.  Name  the  highest  multiple  of  the  number  at  the  head 
of  the  column  in  each  number  in  the  above  columns,  and 
give  the  difference  between  the  multiple  and  the  number 
in  the  column,  thus  for  column  a  ;  18  and  2 ;  9  and  2;  15 
and  1,  etc. 

3.  Divide  the  numbers  in  each  column  by  the  number 
above  the  column,  giving  the  quotient  and  remainder, 
thus  for  column  a  :  6  and  2  over ;  3  and  2  over,  etc. 

4.  Divide  21,487,249  by  6;  459,738,795  by  8;  59,482,395 
by  7  ;  708,718,907  by  9. 


LONG  DIVISION  63 

LONG  DIVISION 

77,  When  all  the  steps  in  division  are  written,  the 
process  is  called  long  division.  Long  division  is  generally 
used  with  divisors  of  two  or  more  places. 

For  the  purpose  of  finding  the  quotient  figure,  all  divi- 
sors are  classified  into  two  general  cases,  as  follows : 

Case  I.  All  divisors  in  which  the  second  figure*  is 
the  same  as  or  less  than  the  first  figure,  as  Q6,  65^  832. 

Case  II.  All  divisors  in  which  the  second  figure  is 
greater  than  the  first  figure,  as  15,  68,  271,  795. 

Write  in  order  all  numbers  from  13  to  100  that  come 
under  Case  I,  as  21,  22;  all  that  come  under  Case  II, 
as  13,  14,  15. 

Case  I 

78.  When  the  second  figure  of  the  divisor  is  the  same  as 
or  less  than  the  first  figure,  a  trial  quotient  figure  may  be 
found,  as  in  the  following : 

Divide  2792  by  65. 

It  will  take  three  places  at  the  left  of 
42      2792  to  contain  65  at  least  once. 
Model  :  65)2792  ^i^P  1-  6  is  contained  in  27  four  times, 

260         with  3  remainder.     Is  5  contained  in  39 
^Qfj      as  many  as  4  times  ?   Yes.   Try  4  as  a  quo- 
tient figure.    Place  4  in  quotient  above  9. 
Step  2.  Multiply  the  divisor  65  by  4,  writing  the  product  below  279. 
.  Step  3.   Subtract  260  from  279. 
Step  4.   Bring  down  the  next  figure  in  the  dividend.     The  new 
dividend  is  192. 

Repeat  Step  1.  6  is  contained  in  19  three  times,  with  1  remainder. 
Is  5  contained  in  12  as  many  as  3  times?  jN"o.  Try  1  less  than  3,  or 
2,  as  a  quotient  figure. 

Complete  the  work,  writing  the  remainder  as  in  short  division. 

*  In  66  regard  6  as  the  first  figure  and  5  as  the  second  figure. 


64  REVIEW   OF  INTEGERS  AND  DECIMALS 

With  divisors  of  Case  I,  the  trial  quotient  figure  found 
as  in  the  model  is  always  the  correct  quotient  figure  when 
the  divisor  contains  two  places,  as  42,  87,  etc. 

It  is  also  the  correct  quotient  figure  when  the  divisor 
contains  three  or  more  places,  whenever  the  second  figure 
of  the  divisor  is  not  contained  in  its  dividend  as  many 
times  as  the  first  figure  is  contained  in  its  dividend. 

With  divisors  of  Case  I,  when  the  first  figure  of  tlie 
divisor  is  contained  in  its  dividend  ten  times,  the  correct 
quotient  figure  is  9,  thus  :  9 

532)52678 

5  is  contained  in  52  ten  times.  Take  9  as  the  quotient 
figure.  It  is  unnecessary  to  test  the  second  figure  of  the 
divisor. 

79.   Written  Exercises. 

Tell  to  which  case  each  divisor  in  the  following  belongs  : 
Solve  only  the  exercises  in  which  the  divisors  are  of 
Case  I. 


1. 

75,679  by  42 

13. 

15,672  by  29 

25. 

57,606  by 

21 

2. 

73,496  by  24 

14. 

71,896  by  83 

26. 

40,000  by 

20 

3. 

12,500  by  64 

15. 

83,678  by  95 

27. 

59,684  by 

84 

4. 

62,847  by  m 

16. 

79,678  by  88 

28. 

85,678  by 

96 

5. 

95,438  by  27 

17. 

53,678  by  61 

29. 

45,672  by 

63 

6. 

18,245  by  29 

18. 

29,678  by  54 

30. 

456,783  by 

15 

7. 

10,000  by  75 

19. 

38,006  by  47 

31. 

648,739  by 

16 

8. 

60,000  by  85 

20. 

47,608  by  96 

32. 

457,820  by 

14 

9. 

35,640  by  44 

21. 

64,896  by  88 

33. 

426,789  by 

13 

10. 

11,045  by  19 

22. 

52,873  by  68 

34. 

480,068  by 

96 

11. 

16,712  by  18 

23. 

49,678  by  79 

35. 

124,530  by 

144 

12. 

27,672  by  28 

24. 

68,368  by  72 

36. 

231,672  by 

772 

LONG  DIVISION  65 

Case  II 

80.  For  the  purpose  of  finding  a  trial  quotient  figure 
that  will  seldom  vary  much  from  the  correct  quotient 
figure,  divisors  of  Case  II  are  classified  into  three  groups. 

Group  a.  When  the  second  figure  of  the  divisor  is  7, 
8,  or  9,  as  17,  18,  19  ;  578,  588,  598,  etc. 

Group  h.  When  the  first  figure  is  more  than  1  and  the 
second  figure  is  3,  4,  5,  or  6,  as  23,  24,  35,  46,  etc. 

Group  c.  When  the  first  figure  is  1  and  tlie  second 
figure  is  3,  4,  5,  or  6,  Le,  13,  14,  15,  16. 

Write  tlie  numbers  from  13  to  100  that  come  under 
Group  a.  Case  II ;  that  come  under  Group  6,  Case  II ; 
that  come  under  Group  c.  Case  II. 

81.  Group  a.  With  divisors  of  Group  a,  the  trial  quotient 
figure  may  be  found  by  using  as  a  divisor  1  more  than  the 
first  figure  of  the  divisor,  as  in  the  following : 

1.  Divide  379,868  by  476. 

Model  : 

79  It  will  take  four  places  to  contain  476  at  least 

4-7fiy^7Q8^       once.     5  (1  more  than  4)  is  contained  in  37  seven 
times.     Try  7  as  a  quotient  figure.     Complete  the 
??£^         division. 
4666 

The  trial  quotient  figure  found  as  in  the  model  will 
sometimes  be  1  less  than  the  correct  quotient  figure. 

2.  Solve  all  exercises  in  Sec.  79  in  which  the  divisors 
are  of  Group  a.  Case  II. 

3.  Write  and  solve  five  exercises  in  division,  using 
divisors  of  Group  a.  Case  II,  and  five  exercises  using 
divisors  of  Case  I. 

MOCL.    &   JONES'S   ESSEN.    OF   AB.  —  6 


66  REVIEW  OF   INTEGERS   AND  DECIMALS 

82.  Group  b.  With  divisors  of  Group  6,  Case  II,  the  trial 
quotient  figure  may  be  found  by  using  as  a  divisor  1  more 
than  the  first  figure  of  the  divisor  and  adding  1  to  the  quo- 
tient, as  in  the  following  : 

1.  Divide  11,678  by  24. 

4 

Tv/r^T.T,r  .         o  i  \-iior7o  I*  will  *^^®  three  places  to  contain  24 

Model:        24)llb<8        -i      .  o  ^i  xi.      ^.x  • 

at  least  once.     3  (1  more  than  2)  is  con- 

*^^  tained  in  11  three  times.      Try  4  as  a 

207         quotient  figure.     Complete  the  division. 

The  trial  quotient  figure  found  as  in  the  model  will 
sometimes  vary  1  from  the  correct  quotient  figure. 

2.  Solve  all  exercises  in  Sec.  79  in  which  the  divisors 
are  of  Group  5,  Case  II. 

3.  Write  and  solve  ten  exercises  in  division,  using  di- 
visors of  Groups  a  and  5,  Case  II. 

4.  Write  and  solve  five  exercises  in  division,  using 
divisors  of  Case  I. 

83.  Group  c.  With  divisors  of  Group  c,  Case  II,  the  trial 
quotient  figure  may  be  found  by  using  2  as  a  divisor,  and 
adding  2  to  the  quotient,  as  in  the  following : 

1.  Divide  12,678  by  142. 

It  will  take  four  places  in  the  divi- 

dend  to  contain  142  at  least  once.     2  is 

Model:         142)12678      contained  in  12  six  times.     Try  8  as  a 
quotient  figure.    Complete  the  division. 

The  trial  quotient  figure  found  as  in  the  model  will 
seldom  vary  more  than  1  from  the  correct  quotient  figure. 

2.  Solve  all  exercises  in  Sec.  79  in  which  the  divisors 
are  of  Group  c,  Case  II. 

3.  Write  and  solve  ten  exercises  in  division,  using 
divisors  of  Case  II. 


LONG  DIVISION  67 

84.  With  1367  as  a  dividend,  find  the  first  trial  quo- 
tient figure,  using  each  of  the  following  as  divisors,  and 
explain  how  each  is  found  :  32,  16,  327,  48,  59,  24,  375, 
698,  166,  426,  276,  149,  137,  161. 

85.  Written  Exercises. 

1.  Write  and  solve  ten  exercises  in  division,  using  as 
divisors  numbers  of  Case  I  between  100  and  1000. 

2.  Write  and  solve  ten  exercises  in  division,  using  as 
divisors  numbers  of  Groups  a,  5,  and  (?,  Case  II,  between 
100  and  1000. 

86.  Written  Exercises. 

1.  How  many  dozen  eggs  at  18^  a  dozen  must  be  sold 
to  pay  for  1  lb.  of  tea  at  60/,  and  1  lb.  of  coffee  at  30/? 

2.  A  man's  yearly  salary  is  f  1860.  Find  his  salary 
per  month.     What  is  the  amount  of  his  salary  per  week? 

3.  If  a  train  travels  at  an  average  rate  of  45  mi.  an 
hour,  in  how  many  hours  will  it  travel  2000  mi.  ? 

4.  A  ton  is  2000  lb.  How  many  pupils  of  your  own 
weight  will  it  take  to  weigh  IT.? 

5.  A  bushel  of  wheat  weighs  60  lb.  How  many 
bushels  will  it  take  to  weigh  IT.? 

6.  The  monthly  rental  of  an  apartment  house  amounted 
to  ^144.  The  average  rental  of  an  apartment  was  $24. 
How  many  apartments  were  there  in  the  house? 

7.  The  total  annual  expenditure  of  the  United  States 
government  for  the  year  ending  June  30, 1905,  amounted  to 
1532,122,762.47.  What  was  the  average  daily  expenditure? 

8.  The  number  of  school  children  in  a  certain  city  is 
1640.  If  the  average  number  of  pupils  to  each  room  is  40, 
how  many  schoolrooms  are  there  in  the  city? 


68  REVIEW  OF  INTEGERS  AND   DECIMALS 

DIVISION  OF  DECIMALS 

87.  1.  How  many  $2  are  there  in  $4?  How  many  2 
tenths  are  there  in  4  tenths?     .4  -j-  .2  =  x? 

2.  How  many  3  qt.  are  there  in  9  qt.  ?  How  many 
3  hundredths  are  there  in  9  hundredths? 

9  qt.  ^3  qt.  =x?         .09 -t- .03  =  a;? 

3.  How  many  times  are  4  yd.  contained  in  8  yd.  ?  4 
tenths  in  8  tenths?  4  hundredths  in  8  hundredths?  4 
thousandths  in  8  thousandths? 

4.  What  is  the  quotient  in  each  of  the  following : 
4)8;  AyS;  .04)708;  .004:)Am?  Prove  the  correct- 
ness of  your  answer  by  multiplying  the  divisor  by  the 
quotient  and  comparing  it  with  the  dividend. 

5.  If  the  divisor  contains  tenths,  tenths  of  the  divi- 
dend may  give  a  whole  number  in  the  quotient. 

3  36 

.4)1.2  .6)21.6 

6.  If  the  divisor  contains  hundredths,  hundredths  of 
the  dividend  may  give  a  whole  number  in  the  quotient. 

9  59  8  54 

.05).45  .04)2.36  .04)34.16 

Place  the  decimal  point  in  the  quotient  above  and  after  the 
figure  in  the  dividend  occupying  the  same  order  as  the  lowest 
order  in  the  divisor.     Divide  as  in  integers. 

,    7.    Divide  21.66  by  6. 

Model  :  As  the  lowest  order  in  the  divisor  is  units, 

6)21.66     place   the  decimal  point  in   the    quotient 

above  and  after  the  figure  occupying  units'  order  in  the  dividend. 

Divide  as  in  integers.  When  the  divisor  is  an  integer,  the  lowest 
order  in  the  divisor  is  units,  and  the  decimal  point  in  the  quotient  is 
directly  above  the  decimal  point  in  the  dividend. 


LONG  DIVISION  69 

a  Divide  43.38  by  .8. 

Model-       8^4S  S8         ^^  *^®  lowest   order  in   the  divisor  is 
^      *  tenths,  place  the  decimal  point  above  and 

after  the  figure  occupying  tenths*  order  in  the  dividend.     Divide. 

9.    Divide  2.4  by  .006. 

Model  ;         006^2  400  ^'^  *^®  lowest  order  in  the  divisor  is 

thousandths,  supply  two  ciphers  in  the 
dividend  to  make  the  lowest  order  thousandths.  Place  the  decimal 
point  above  and  after  the  figure  in  the  dividend  occupying  thou- 
sandths' order.     Divide. 

10.  With  45.06  as  a  divisor,  the  decimal  point  will  be 
placed  in  the  quotient  above  and  after  the  figure  in  the 
dividend  occupying  hundredths'  order.  State  where  the 
decimal  point  should  be  placed  with  each  of  the  following 
as  divisors:  5.05;  67;  3.15;  3.1416;  .008;  26.1; 
.0045;  50;  .05;  2150.42;  6. 

88.  Arrange  as  in  the  models  and  fix  the  decimal  points 
in  the  quotients  : 

a  bed 

1.  .05-5-2.5  .04-^.002      180 -4-. 006  3^4 

2.  .6-^.3  2.5-^50  36-^750         20^50 

3.  1.44 -f- .12       .0048 -5- .6  .27-^-3  1.75 -f- .025 

4.  2.7 -f- 9  3.6-^.12       .007^3.5       .075-^2.5 

5.  .48-1-8  .16^20  14 -.007      120 -5- .004 

6.  .1--.005         2.4-5-12        22.5 -T- .15        4.8 -5- .0012 

7.  .024^.8  .012-^.03  2-f-lO        .065-^3.25 

8.  3.6-^.006       .005 -4- .1  45-^-90  64 -f- .0008 

9.  1-4-.045  4-4.56      100^.1  .75-^.6 
10.       10-?- .01           .01^10        .001 -5- .01        101 -i- 1.01 


70  REVIEW  OF  INTEGERS  AND  DECIMALS 

89,   1.    Divide  75.51  by  60.4,  and  carry  the  result  to 
two  decimal  places. 

1.25  + 
Model:      60.4)75.51  ^^^  quotient,  carried  to  two  deci- 

'  mal  places,  is  1 .25.     The  sign  (  +  )  is 

^  ^  placed  after  the  quotient  to  indicate 

15  11  that  the  division  is  not  exact. 

12  08 
3  03 
Arrange  as  in  the  model,  fix  the  decimal  point,  estimate 
the  result,  then  divide.     When  the  division  is  inexact, 
carry  the  quotient  to  three  decimal  places. 


2. 

6.25-25 

12. 

500-^.005 

3. 

.1728 -^  .12 

13. 

512.16-^64.02 

4. 

720.405  ^  3.15 

14. 

68.045-^-42,125 

5. 

250-^.75 

15. 

12.5  H- .0375 

6. 

1210.605-^6.05 

16. 

1000^.875 

7. 

.0045 -f- .045 

17. 

12.75^3.1416 

8. 

37.806-^8.7 

18. 

458,766  -f-  2150.42 

9. 

48.312-^3.1416 

19. 

8.05  -^  40.25 

10. 

1000-^.0025 

20. 

8790  ^  2150.42 

11. 

.1224-^2.04 

21. 

100 -.125 

22. 

Read  the  decimals  in  the  above  exercises. 

23.  When  a  number  is  divided  by  .4,  is  the  quotient 
greater  or  less  than  the  number  ? 

24.  When   a   number   is  divided   by  .2,  the   quotient 
obtained  is  how  many  times  the  dividend? 

25.  Estimate  the  quotient  of  12  divided  by  each:  .1, 
.25,  .5,  .4. 

26.  State  a  short  method  of  dividing  by  10 ;  by  100 ; 
by  1000 ;  by  .1 ;  by  .01 ;  by  .001. 


REVIEW  71 

REVIEW 

90.  1.    Solve  exercises  in  Sec.  88. 

2.  Write  five   exercises   in  division  of  decimals,  and 
solve  each. 

3.  Write  five  exercises  in  multiplication  of   decimals, 
and  solve  each. 

4.  Write  five  exercises  in  Case  I  in  long  division,  and 
solve  each. 

5.  Write  five  exercises  in  Group  a  of  Case  II  in  long 
division,  and  solve  each. 

6.  Write  five  exercises  in  Group  h  of  Case  II  in  long 
division,  and  solve  each. 

7.  Write  five  exercises  in  Group  c  of  Case  II  in  long 
division,  and  solve  each. 

8.  Write  five  columns  in  addition,  and  add  each  as  in- 
dicated in  Sec.  73. 

91.  Add  the  following : 


1. 

2. 

3. 

$786.45 

$578.04 

%   16.45 

97.08 

35.16 

8.12 

300.90 

900. 

947. 

7.87 

80.47 

32.76 

46.59 

570.09 

6.58 

807.98 

98.17 

300. 

345.56 

315.40 

97.26 

96. 

9.98 

1.95 

4.75 

405.56 

647.15 

400. 

58.08 

45. 

57.09 

930. 

780.35 

815.35 

40.76 

46.27 

72  REVIEW   OF  INTEGERS  AND  DECIMALS 

92.  1.  Give  the  number  of  pints  in  a  quart ;  of  quarts 
in  a  gallon ;  of  pints  in  a  gallon. 

2.  How  many  ounces  are  there  in  a  pound  of  siigar? 
in  5  lb.  ?     How  many  pounds  are  there  in  a  ton? 

3.  There  are  2000  lb.  in  a  short  ton,  and  2240  lb.  in 
a  lonff  ton.  A  company  imported  11,200,000  lb.  of  coal, 
paying  for  it  by  the  long  ton.  The  company  sold  the 
coal  by  the  short  ton.  How  many  more  tons  did  it  sell 
than  it  imported? 

A  long  ton  (2240  lb.)  is  used  sometimes  in  weighing  coal  and  in 
weighing  certain  materials  imported  into  the  United  States. 

4.  In  dry  measure  2  pints  are  1  quart,  8  quarts  are 
1  peck,  and  4  pecks  are  1  bushel.  How  many  quarts  are 
there  in  1  bu.  ?  in  1  pk.  and  3  qt.  ?  in  1  bu.  2  pk.? 

5.  How  many  months  are  there  in  1  yr.  ?  in  7  yr.? 
How  many  days  are  there  in  1  yr.  ?  in  1  leap  year? 

6.  The  depth  of  the  sea  is  measured  in  fathoms,  A 
fathom  is  6  ft.     Express  1728  ft.  in  fathoms. 

7.  The  circumference  of  a  circle  is  3^  (3.1416)  times 
its  diameter.  Show  that  this  is  correct  by  comparing  the 
diameter  and  the  circumference  of  some  circle  (top  of 
barrel,  pail,  stovepipe,  etc.). 

8.  A  denominate  number  is  a  concrete  number  in 
which  the  unit  of  measure  has  been  established  by  law 
or  custom,  as  4  ft.,  12  gal.,  etc.  Such  expressions  as  10  ft. 
6  in.,  2  yr.  7  mo.  6  da.,  etc.,  are  called  compound  denomi- 
nate numbers. 

Tables  of  denominate  numbers  are  found  on  pp.  312-319. 

9.  Mr.  Davis's  expenses  for  January,  1907,  were 
$121.45.     What  were  his  average  daily  expenses? 


REVIEW  73 

93.  1.  The  exports  for  the  first  ten  months  of  1906 
amounted  to  $1,425,184,757,  while  the  exports  for  the 
corresponding  period  in  1905  amounted  to  $1,256,924,354. 
At  the  same  rate  of  increase,  by  how  much  would  the 
exports  for  1906  exceed  the  exports  for  1905  ? 

2.  The  imports  for  the  first  ten  months  of  1906 
amounted  to  $1,066,462,295,  while  the  imports  for  the 
first  ten  months  of  1905  amounted  to  $979,917,437.  At 
the  same  rate  of  increase,  by  how  much  should  the  im- 
ports for  1906  exceed  the  imports  for  1905? 

3.  The  population  of  Massachusetts  was  2,805,346  in 
1900.  The  area  of  Massachusetts  is  8315  sq.  mi.  Find 
the  average  population  for  each  square  mile. 

4.  The  area  of  Texas  is  265,780  sq.  mi.  and  of  Iowa  is 
56,025  sq.  mi.  How  many  states  of  the  size  of  Iowa  can 
be  made  of  Texas? 

5.  Four  places,  A,  B,  C,  and  D,  are  located  on  a  line 
running  due  east  and  west.  B  is  16  mi.  east  of  A,  C  is 
12  mi.  west  of  A,  and  D  is  8  mi.  west  of  C.  How  far 
apart  are  B  and  C?  A  and  D?  B  and  D?  (Draw 
a  diagram.) 

6.  Mr.  Wright  of  Chicago  is  employed  by  a  wholesale 
house  and  receives  $125  per  month  and  necessary  expenses 
while  traveling.  During  the  month  of  January  Mr. 
Wright  paid  $69  for  railroad  fare,  $86  for  hotel  bills, 
and  $18.65  for  other  expenses.  How  much  did  the  com- 
pany owe  Mr.  Wright  for  the  month,  including  salary? 

7.  Charles  deposited  $2.75  in  a  savings  bank  on  Oct. 
15,  and  $3.45  on  Oct.  23.  He  drew  out  $4.10  on  Oct.  29. 
He  deposited  $4.80  on  Dec.  1.  How  much  had  he  then 
in  the  bank? 


74 


REVIEW  OF  INTEGERS  AND  DECIMALS 


94.  Reading  a  Railroad  Time  Table. 

SAN  FRANCISCO  — LOS  ANGELES 


20 

Shore 

Line 

Limited 

22 

The 

Coaster 

18-8 
Los 
Angeles 
Passen- 
ger 

10 

Sunset 
Ex- 
press 

ai 

8T1TI0I8 

17 
San 
Fran- 
cisco 
Passen- 
ger 

9 

Sanset 
Ex- 
press 

19 

Shore 

Line 

Umited 

21 

The 
Coaster 

READ    DOWN 

0 

Lv. 

SAN  FRANCISCO  Ar. 

READ   UP 

8.00 

8.80 

3.15 

8.00 

9.16 

10.15 

9.30 

11.45 

9.25 

9.55 

4.45 

9.30 

51 

Lv. 

SAN  JOSE         Lv. 

7.86 

8.45 

8.05 

10.15 

6.19 

8.15 

4.85 

7.30 

871 

Lv.  SANTA  BARBARA  Lv. 

8,20 

11.00 

11.15 

12.10 

9.30 

11.45 

8.45 

11.00 

475 

Ar. 

LOS  ANGELES     Lv. 

4.00 

7.30 

8.00 

8.80 

Light-face  figures,  a.m.  ;  dark-face,  p.m. 

95.   Answer  the  following  from  the  above  table  : 

1.  How  many  passenger  trains  leave  San  Francisco  for 
Los  Angeles  each  day  over  this  route  ?  How  many  leave 
Los  Angeles  for  San  Francisco  ? 

2.  What  is  the  distance  from  San  Francisco  to  Los 
Angeles  ? 

3.  What  is  denoted  by  the  light-face  figures  ?  by  the 
dark-face  figures  ? 

4.  Which  is  the  first  train  in  the  morning  from  San 
Francisco  to  Los  Angeles?  from  Los  Angeles  to  San 
Francisco  ? 

5.  How  many  hours  does  it  take  each  train  to  make 
the  run  ?     Which  are  the  fastest  trains? 

6.  Find  the  average  number  of  miles  per  hour  of  train 
No.  20,  of  train  No.  22,  and  of  the  Sunset  Express,  on 
the  run  from  San  Francisco  to  Los  Angeles,  and  on  the 
run  from  Los  Angeles  to  San  Francisco. 

7.  Find  the  distance  from  Los  Angeles  to  Santa 
Barbara;  from  Los  Angeles  to  San  Jose;  from  San  Jose 
to  Santa  Barbara. 


REVIEW 


75 


96.   Reading  a  Meter. 

The  amount  of  water,  gas,  and  electricity  consumed 
is  usually  measured  by  instruments  called  meters.  These 
instruments  are  furnished  with  dials,  on  which  the 
amounts  consumed  are  indicated  in  the  decimal  scale,  as 
shown  in  the  picture. 


97.   Dials  of  a  Gas  Meter. 


CUBIC 


The  unit  dial  at  the  top  is  used  for  testing  the  meter. 

For  every  100  cu.  ft.  of  gas  that  passes  through  the 
meter,  the  hand  on  the  first  (right-hand)  dial  moves  over 
one  of  the  divisions,  as  from  0  to  1  ;  for  every  1000  cu.  ft. 
consumed,  it  makes  a  complete  revolution,  the  hand  on 
the  second  dial  moves  over  one  division,  and  the  hand  on 
the  third  dial  moves  over  -^^  of  one  division. 

Ten  revolutions  of  the  hand  on  any  dial  produce  one 
revolution  of  the  hand  on  the  dial  of  the  next  higher 
order. 

The  first  dial  is  now  recording  300  cu.  ft.  How  much 
is  the  second  dial  recording  ?  How  much  are  the  three 
dials  recording  ?  The  dials  should  be  read  from  left  to 
right  as  you  would  read  a  number,  thus  :  68,300  cu.  ft. 

The  cost  of  the  gas  would  be  stated  for  each  1000  cu.  ft. 


76  REVIEW  OF   INTEGERS   AND   DECIMALS 

MEASUREMENT  OF  LENGTH 

98.  1.  Length  and  distance  are  commonly  measured  in 
inches,  feet,  yards,  rods,  and  miles.  The  yard  is  the  stand- 
ard unit  of  length.     The  other  units  are  derived  from  it. 

2.  Draw  on  the  blackboard  a  line  1  in.  long.  Draw 
a  line  1  ft.  long.  Draw  a  line  1  yd.  long.  Using  a 
yard  stick,  test  the  correctness  of  your  drawings.  Prac- 
tice drawing  these  lines  until  you  can  estimate  an  inch,  a 
foot,  and  a  yard  without  much  error. 

3.  Estimate  in  inches  the  length  and  the  width  of 
each  :  your  desk  top ;  your  book  cover ;  a  window  pane. 

4.  Estimate  in  feet  the  length  and  the  width  of  each : 
your  schoolroom  ;  the  blackboard ;  the  window ;  the  door. 

5.  Estimate  the  length  of  your  room  in  yards  ;  of  your 
school  yard  ;  of  the  blackboard. 

6.  A  rod  is  16 J  ft.,  or  5|  yd.  Measure  off  a  rod  on  the 
floor  of  your  schoolroom  or  on  the  school  yard.  Estimate 
the  length  and  width  of  your  school  yard  in  rods. 

7.  Determine  some  place  that  is  1  mi.  from  your 
schoolhouse. 

99.  Table  of  Linear  Measure. 

12  inches  (in.  or  "^  =  1  foot  (ft.  or  ') 
3  ft.  =1  yard  (yd.) 

5^  yd.,  or  16J  ft.     =  1  rod  (rd.) 
320  rd.,  or  5280  ft.     =  1  mile  (mi.) 

1.  How  many  feet  are  there  in  3  mi.  ?  in  5  mi.  ? 

2.  How  many  rods  are  there  in  2  mi.  ?  in  J  mi.?  in  \ 
mi.? 

3.  Change  to  rods  :  \  mi.,  \  mi.,  J  mi. 

4.  How  many  inches  are  there  in  1  yd.  6  in.? 


MEASUREMENT  OF  LENGTH  77 

5.  The  lengths  of  three  pieces  of  blackboard  in  a  school- 
room were  measured  by  the  pupils  and  found  to  be  18  ft. 
6  in.,  14  ft.  9  in.,  and  6  ft.  4  in.,  respectively.  Find  the 
combined  length  of  the  three  boards. 

Model  : 
18  ft.  6  in.  4  in.  and  9  in.  and  6  in.  are  19  in.,  or  1  ft.  and  7 

14  ft.  9  in.  ^^-     Write  7  in.  in  the  answer  as  shown  in  the 

6  ft.  4  in.  ^odel,  and  carry  1  ft.  to  the  column  of  feet. 

39  ft.  7  in. 

6.  Find  the  combined  length  of  the  blackboards  in 
your  schoolroom. 

7.  Find  the  distance  around  your  schoolroom. 

8.  From  8  ft.  4  in.  subtract  4  ft.  10  in. 

Model  :  ^^  ^^  ^"-  ^^®  more  than  4  in.,  the  sum  of  10  in. 
8  ft      4  in  ^^^  ^^®  number  of  inches  in  the  answer  is  1  ft.  4 
A  jfi^    1  r\  '  ill-     Subtract  thus  :  10  in.  and  2  in.  are  1  f t. :  2  in. 
4  ft.  10  in.  ,   .   .            ^  .       ,nr  .^    o  .     .     ^. 
. —  and  4  in.  are  6  m.     Write  6  in.  in  the  answer  as 

o  It.  o  m.  s]iown  in  the  model.  Carry  1  ft.  to  4  ft.,  making 
5  ft.     5  ft.  and  3  ft.  are  8  ft. 

9.  From  a  board  9  ft.  6  in.  long  a  carpenter  sawed  a 
shelf  3  ft.  10  in.  long.  How  long  was  the  piece  of  board 
that  was  left  ? 

10.  From  a  piece  of  cloth  4  yd.  8  in.  long  a  woman  cut 
a  piece  1  yd.  9  in.  long.  How  long  was  the  piece  of  cloth 
that  was  left  ? 

11.  Find  how  much  longer  the  length  of  your  school- 
room is  than  its  width. 

12.  On  Jan.  1,  1903,  a  boy's  height  was  4  ft.  7  in.,  and 
on  Jan.  1,  1906,  it  was  5  ft.  2  in.  How  much  taller  was 
he  on  the  second  date  ? 

13.  How  many  feet  are  there  in  1  mile?  How  many 
yards  are  there  in  1  mile  ? 


78  REVIEW  OF  INTEGERS  AND  DECIMALS 

BIEASUREMENT  OF  SURFACES 

100.  1.  The  number  of  square  units  in  any  surface 
is  called  its  area. 

2.  The  area  of  surfaces  is  commonly  measured  in 
square  inches,  square  feet,  square  yards,  square  rods,  acres, 
or  square  miles. 

3.  Using  a  ruler,  draw  on  the  board  a  square  whose 
side  is  1  foot.  This  is  called  a  square  foot.  A  square 
foot  is  a  square  whose  side  is  1  foot. 

4.  Using  a  ruler,  draw  upon  the  board  a  square  whose 
side  is  1  inch.  This  is  called  a  square  inch.  A  square 
inch  is  a  square  whose  side  is  1  inch. 

5.  Divide  a  square  foot  into  square  inches.  How  many 
square  inches  are  there  in  1  square  foot? 

6.  Using  a  yard  stick,  draw  a  square  whose  side  is  1 
yard.  What  is  this  square  called?  Divide  a  square  yard 
into  square  feet.  How  many  square  feet  are  there  in  1 
square  yard? 

7.  What  is  the  purpose  of  having  several  different 
units  for  measuring  length  and  area?  In  what  unit 
should  you  express  the  area  of  the  cover  of  this  book? 
of  the  top  of  your  desk  ?  of  the  surfaces  of  the  walls  in 
your  schoolroom  ? 

8.  Mark  out  a  square  rod  on  the  school  yard. 

101.  Table  of  Square  Measure. 

144  square  inches  (sq.  in.)  =  1  square  foot  (sq.  ft.) 

9  sq.  ft.  =  1  square  yard  (sq.  yd.) 

30 J  sq.  yd.  =  1  square  rod  (sq.  rd.) 

160  sq.  rd.  =  1  acre  (A.) 

640  A.  =1  square  mile  (sq.  mi.) 


ANGLES  19 


ANGLES 


RightAngle 


102.  1.  The  difference  in  direction  of  two  lines  is  called 
an  angle.  The  name  angle  is  used  also  to  denote  the 
opening  between  two  lines  that  meet. 

2.  When  two  straight  lines  meet 
and    form    two    equal    angles,    the 
angles  are  called  right  angles,  and 
the  lines  are  said  to  be  perpendicular    RighfAngle 
to  each  other. 

3.  The  lines  that  form  an  angle  are  called  the  sides  of 
the  angle. 

4.  Angles  whose  sides  are  not  perpendicular  to  each 
other  are  called  oblique  angles.  Oblique  angles  are  either 
acute  or  obtuse.  /\  \      / 

5.  An  angle  that  is  less  than  a        /     \  v 
right  angle  is  called  an  acute  angle.              Acute  angles. 

6.  An  angle  that  is  greater  than  a 

right  angle  but  less  than  two   right       /^"^     ^^^^^ 
angles  is  called  an  obtuse  angle.  Obtuse  angles. 

7.  Draw  a  right  angle  ;  an  obtuse  angle  ;  an  acute 
angle. 

8.  Stand  your  pencil  upon  the  top  of  your  desk,  per- 
pendicular to  the  desk  top.  What  kind  of  angles  are 
formed  by  the  pencil  and  the  desk  top  ? 

9.  Keeping  the  pencil  resting  at  the  same  point  on 
the  top  of  the  desk,  move  the  top  of  the  pencil  to  the  left. 
Does  the  pencil  now  form  any  right  angles  with  the 
desk  top  ?  any  obtuse  angles  ?  any  acute  angles  ? 

10.  Point  to  surfaces  in  your  schoolroom  that  meet  at 
right  angles.  Are  there  any  that  meet  so  as  to  form 
obtuse  angles  ?  acute  angles  ? 


80  REVIEW  OF  INTEGERS  AND  DECIMALS 

RECTANGLES 

103.   1.    Lines  that  extend  in  the  same  direction  and 
are  everywhere  the  same  distance  apart  are 

— called  parallel  lines.     If  parallel  lines  are 

Parallel  Lines,  extended,  will  they  ever  meet  ? 

2.  Point  to  two  surfaces  in  the  schoolroom  that  are 
parallel  to  each  other  ;  that  are  perpendicular  to  each 
other  ;  that  meet  at  right  angles.  Are  there  any  sur- 
faces in  your  schoolroom  that  meet  at  obtuse  or  acute 
angles  ? 

3.  A  figure  having  four  straight  sides  and  four  right 

angles  is  called  a  rec- 
tangle. Find  by  drawing 
figures  how  many  sides  a 
figure  must  have  in  order 

Rectanglks.  t^a^  all  Its  angles  may  be 

right  angles. 

4.  A  rectangle  whose  sides  are  equal  is  called  a 
square. 

5.  What  is  a  square  inch  ?  Draw  a  square  inch; 
a  square  foot.     What  is  a  square  yard  ?  a  square  mile  ? 

6.  All  rectangles  are  either  square  or  oblong.  Point 
to  surfaces  in  the  schoolroom  that  are  rectangles. 

7.  Draw  a  2-inch  line.  Build  a  square  upon  it.  This 
is  a  2-inch  square.  Divide  it  into  square  inches.  How 
many  square  inches  are  there  in  a  2-inch  square  ? 

8.  Draw  a  3-inch  square.  Divide  it  into  square 
inches.  How  many  square  inches  are  there  in  a  8-inch 
square  ? 

9.  Draw  a  rectangle  2  in.  wide  and  8  in.  long.  Di- 
vide it  into  square  inches. 


RECTANGLES 


81 


■ 

■ 

• 
« 

104.  Area  of  Rectangles. 

1.  Draw  a  rectangle  6  in. 
long  and  1  in.  wide.  Divide 
it  into  square  inches. 

A  rectangle  6  in.  long  and  1  in.  wide  contains  6  sq.  in. 

2.  Draw  a  rectangle  6  in.  long  and  3  in.  wide.  Divide 
it  into  square  inches. 

As  both  dimensions  are 
given  in  inches,  the  unit  in 
which  the  area  of  the  rectangle 
is  to  be  expressed  is  1  sq.  in. 
In  a  rectangle  6  in.  long  and 
1  in.  wide  there  are  6  times  1  sq.  in.,  or  6  sq.  in.  Since 
the  given  rectangle  is  3  in.  wide,  it  contains  3  times 
6  sq.  in.,  or  18  sq.  in. 

Or,  a  rectangle  3  in.  long  and  1  in.  wide  contains  3 
times  1  sq.  in.,  or  3  sq.  in.  Since  the  given  rectangle 
contains  6  such  rectangles,  its  area  is  6  times  3  sq.  in.,  or 
18  sq.  in. 

3.  The  number  of  square  inches  in  the  given  rectangle 
may  be  found  by  multiplying  6  by  3.  6  times  3  is  18,  the 
number  of  square  inches  in  the  rectangle.  Never  multiply 
inches  by  inches.     Why  ? 

105.  The  following  are  dimensions  of  rectangles. 
State  the  unit  in  which  the  area  of  each  is  to  be  found. 
Represent  1-4  by  drawings.     Give  the  area  of  each. 


1.  8  in.  by  5  in. 

2.  12  yd.  by  9  yd. 

3.  8  ft.  by  6  ft. 

4.  4  ft.  by  3  ft. 

5.  14  ft.  by  10  ft. 


6.  20  ft.  by  15  ft. 

7.  20  rd.  by  10  rd. 

8.  40  rd.  by  40  rd. 

9.  6  mi.  by  J  mi. 
10.  25  ft.  by  100  ft. 


82  REVIEW  OF  INTEGERS  AND  DECIMALS 

106,  1.  Compare  the  size  of  a  2-inch  square  with  2 
square  inches  ;  of  a  3-inch  square  with  3  square  inches. 

2.  A  4-inch  square  is  how  many  times  4  square  inches  ? 
Compare  a  5-inch  square  with  5  square  inches. 

3.  Draw  an  inch  square.  Divide  it  into  4  equal 
squares.  How  long  is  the  side  of  a  square  that  contains 
one  quarter  of  a  square  inch  ? 

4.  Draw  a  square  that  contains  one  sixteenth  of  a 
square  inch. 

5.  Draw  a  rectangle  containing  18  square  inches, 
making  it  6  inches  long ;  9  inches  long. 

6.  Which  is  greater,  an  inch  square  or  a  square  inch  ? 
a  half-inch  square  or  one  half  of  a  square  inch?  Show 
by  drawing. 

7.  How  long  is  the  perimeter  of  a  rectangle  6  ft.  long 
and  4  ft.  wide?     (Perimeter  means  distance  around.) 

8.  How  long  is  the  perimeter  of  a  9-inch  square  ?  of  a 
4-inch  square  ?  of  a  square  inch  ? 

9.  How  wide  is  a  rectangle  that  is  8  in.  long  and  con- 
tains 8  sq.  in.  ?  16  sq.  in.  ?  24  sq.  in.  ?  32  sq.  in.  ? 

10.  By  what  number  must  T  sq.  in.  be  multiplied  to 
give  28  sq.  in.  ?  If  the  area  of  a  rectangle  is  28  sq.  in. 
and  its  length  is  7  in.,  how  wide  is  the  rectangle  ? 

11.  When  the  area  of  a  rectangle  and  one  dimension 
are  given,  how  may  the  other  dimension  be  found  ?  Illus- 
trate with  several  examples. 

12.  How  many  dimensions  has  a  rectangle  ? 

13.  A  surface  that  has  the  same  direction  throughout, 
as  the  surface  of  a  blackboard,  a  window  pane,  etc.,  is 
called  a  plane  surface.  The  surface  of  a  globe  is  not  a 
plane  surface.     Why  ? 


RECTANGLES  83 

107.  In  each  of  the  following  the  area  of  a  rectangle 
and  one  dimension  are  given.     Find  the  other  dimension  : 

1.  Area,  20  sq.  ft.,  length,  5  ft. 

2.  Area,  48  sq.  yd.,  width,  6  yd. 

3.  Area,  100  sq.  in.,  length,  10  in. 

4.  Length,  45  ft.,  area,  900  sq.  ft. 

5.  Width,  50  ft.,  area,  6500  sq.  ft. 

6.  Area,  1728  sq.  in.,  length,  144  in. 

108.  1.    Find  the  area  of  a  garden  10  rd.  long  and 
8  rd.  wide. 

2.  Find  the  number  of  acres  in  a  field  40  rd.  by  20  rd. 

3.  At  $85  an  acre,  find  the  value  of  a  farm  80  rd.  by 
40  rd. ;  160  rd.  by  80  rd. 

4.  A  farm  containing  80  acres  is  80  rd.  wide.     How 
long  is  it  ? 

5.  How  long  is  a  10-acre  field,  if  its  width  is  40  rd.? 
20  rd.? 

6.  How  long  is  a  20-acre  field,  if  its  width  is  40  rd.  ? 
20  rd.  ? 

109.  1.    Find  the  number  of  square  feet  of  window 
space  in  your  schoolroom. 

2.  Estimate  the  number  of  square  inches  in  the  cover 
of  this  book.     Test  the  correctness  of  your  estimate. 

3.  Estimate  the  area  of  the  floor  of  your  schoolroom  in 
square  feet.     Test  the  correctness  of  your  estimate. 

4.  Estimate  the  number  of  square  rods  in  your  play- 
ground.    Test  the  correctness  of  your  estimate. 

5.  Are  there  as  many  as  60   sq.  yd.  of  surface  in  the 
ceiling  of  your  schoolroom?     Test  your  answer. 


84  REVIEW  OF  INTEGERS  AND   DECIMALS 

CUBIC  MEASURE 

llO,  1.  Describe  a  rectangle ;  a  square ;  an  oblong. 
Draw  each. 

2.  How  many  dimensions  has  a  rectangle  ?  a  plane 
surface  ? 

3.  How  many  dimensions  has  a  book  ?  a  block  ?  a  box  ? 

4.  Any  object  that  has  length,  breadth,  and  thickness 
is  called  a  solid.     Is  a  book  a  solid  ?     Name  other  solids. 

5.  A  solid  having  six  rectangular  faces  is  called  a  rec- 
tangular solid.  Is  a  brick  a  rectangular  solid?  Isame 
objects  which  are  rectangular  solids. 

6.  A  solid  having  six  equal  square  surfaces  is  called  a 
cube.     How  many  edges  has  a  cube? 

^  7.   A  cube  whose  faces  are  each  a  foot 

llHli^       square  is  called  a  cubic  foot.     Describe  a 

^^B  ill       cubic  inch ;  a  cubic  yard. 

8.    How  many  inch  cubes  will  form   a 

solid  12  in.  long,  1  in.  wide,  and  1  in.  thick? 

12  in.  long,  12  in.  wide,  and  1  in.  thick? 

12  in.  long,  12  in.  wide,  and  2  in.  thick? 

12  in.  long,  12  in.  wide,  and  3  in.  thick? 

12  in.  long,  12  in.  wide,  and  6  in.  thick? 

12  in.  loner,  12  in.  wide,  and  12  in.  thick? 
Rectangular  ° 

Solids.  9.  What  name  may  be  given  a  solid 
formed  by  placing  inch  cubes  12  deep  on  a  surface  1  ft. 
square  ?     There  are cu.  in.  in  1  cu.  ft. 

10.  How  many  foot  cubes  will  form  a  solid  3  ft.  long, 
3  ft.  wide,  and  1  ft.  thick  ?  3  ft.  long,  3  ft.  wide,  and  3  ft. 
thick? 

11.  How  many  cubic  feet  are  there  in  a  rectangular 
solid  8  ft.  long,  4  ft.  wide,  and  4  ft.  thick? 


CUBIC   MEASURE  85 

As  the  dimensions  are  all  expressed  in  feet,  the  cubic 
contents  will  be  found  in  cubic  feet.  In  a  rectangular 
solid  having  the  same  base,  8  ft.  by  4  ft.,  but  only  1  f t .  in 
height,  there  are  1  cu.  ft.  x  8  x  4,  or  32  cu.  ft.  Since  the 
given  rectangular  solid  is  4  ft.  thick,  it  contains  4  times 
32  cu.  ft.,  or  128  cu.  ft.    1  cu.  ft.  x  8  x  4  x  4  =  128  cu.  ft. 

When  all  the  dimensions  of  a  rectangular  solid  are  ex- 
pressed in  like  units,  the  contents  of  the  solid  may  be 
found  by  multiplying  the  number  of  units  in  the  length  by 
the  number  of  the  units  in  the  width  and  the  product  by 
the  number  of  units  in  the  thickness,  and  calling  the  result 
cubic  units  of  the  given  dimension.  Thus,  the  number  of 
cubic  feet  in  a  rectangular  solid  8  ft.  by  4  ft.  by  4  ft.  is 
8  X  4  X  4,  or  128. 

12.  A  pile  of  wood  8  ft.  long,  4  ft.  wide,  and  4  ft.  high 
is  called  a  cord  of  wood.  How  many  cubic  feet  are  there 
in  a  cord  of  wood  ? 

13.  The  number  of  cubic  units  in  a  solid  is  called  its 
volumCo 

111.   Table  of  Cubic  Measure. 

1728  cubic  inches  (cu.  in.)  =  1  cubic  foot  (cu.  ft.) 
27  cu.  ft.  =  1  cubic  yard  (cu.  yd.) 
128  cu.  ft.  =  1  cord  of  wood  (cd.) 

Find  the  volume  of  rectangular  solids  of  the  following 
dimensions : 

1.  6'   x4^  x2^  4.    12^'  X    9''x    8'' 

2.  10'  X  8'   X  5'  5.   14"  X  12^'  X  10'' 

3.  9"x6"x4"  6.   24'  xl8'  x  16' 

7.    Find  the  number  of  cubic  feet  of  air  in  a  room  14' 
by  12'  by  9'. 


86  REVIEW  OF  INTEGERS   AND  DECIMALS 

8.  If  each  pupil  requires  35  cu.  ft.  of  fresh  air  per 
minute,  how  many  cubic  feet  of  fresh  air  per  minute  will 
50  pupils  require  ? 

9.  Find  the  number  of  cubic  feet  of  air  in  a  school- 
room 32'  X  24'  X  12'. 

10.  A  watering  trough  is  12  ft.  long,  2  ft.  wide,  and 
2  ft.  deep.  How  many  gallons  will  it  contain  ?  (231  cu. 
in.  =  1  gal.) 

11.  Find  the  number  of  measured  bushels  in  a  bin  6  ft. 
by  4  ft.  by  4  ft.     (2150.42  cu.  in.  =  1  bu.) 

12.  The  length  of  a  rectangular  solid  is  8  ft.,  and  its 
width  is  5  ft.  If  its  volume  is  160  cu.  ft.,  what  is  its 
thickness  ? 

13.  How  many  cubic  yards  of  dirt  must  be  removed  in 
excavating  a  cellar  36  ft.  long,  24  ft.  wide,  and  8  ft.  deep  ? 
Find  the  cost  of  making  this  excavation  at  30^  per  cubic 
yard. 

14.  How  many  cubic  yards  of  dirt  must  be  removed  in 
digging  a  trench  for  a  sewer,  if  the  trench  is  3  ft.  wide, 
6  ft.  deep,  and  120  ft.  long  ? 

15.  A  contractor's  bid  for  excavating  a  basement  60  ft. 
by  36  ft.  and  9  ft.  in  depth  is  $216.  How  much  is  this 
per  cubic  yard  ? 

16.  How  many  cords  of  wood  in  a  pile  24  ft.  long, 
4  ft.  wide,  and  4  ft.  high  ? 

17.  How  many  cords  of  wood  in  a  pile  8  ft.  long,  2  ft. 
wide,  and  4  ft.  high  ? 

18.  At  $6  per  cord,  find  the  cost  of  a  pile  of  wood 
20  ft.  long,  4  ft.  wide,  and  6  ft.  high. 

19.  State  how  you  would  find  the  capacity  of  a  box  car. 


DIYISIBILITY  87 

DIVISIBILITY  OF  NUMBERS 

112.  1.  Count  by  2's  to  20.  Numbers  that  are  ex- 
actly divisible  by  2  are  called  even  numbers.  All  numbers 
ending  in  0,  2,  4,  6,  and  8  are  even  numbers.  Name  the 
even  numbers  from  40  to  60. 

2.  Numbers  that  are  not  exactly  divisible  by  2  are 
called  odd  numbers.     Name  the  odd  numbers  from  20  to  40. 

3.  Which  of  the  following  are  even  numbers :  18,  27, 
31,46,50,65,123,2456? 

4.  Some  numbers  are  not  exactly  divisible  by  any 
whole  number  except  themselves  and  1.  Such  numbers 
are  called  prime  numbers.  Write  the  prime  numbers 
below  80.  Numbers  that  are  exactly  divisible  by  whole 
numbers  other  than  themselves  and  1  are  called  composite 
numbers. 

5.  The  factors  of  15  are  3  and  5.  These  two  numbers 
when  multiplied  together  give  15.  Some  numbers  have 
many  factors.  2,  3,  4,  6,  8,  and  12  are  each  a  factor  of  24. 
By  what  must  each  be  multiplied  to  give  24  ? 

A  number  which  when  multiplied  by  another  number 
makes  a  given  number  is  called  a  factor  of  the  given 
number. 

6.  Name  the  factors  of  6  ;  of  12  ;  of  16  ;  of  30  ;  of 
36.     Has  a  prime  number  factors  ? 

7.  The  prime  factors  of  a  number  are  the  prime  num- 
bers which  when  multiplied  together  make  the  number. 
The  prime  factors  of  24  are  2,  2,  3,  2. 

8.  Name  the  prime  factors  of  18  ;  of  20  ;  of  36. 

9.  A  factor  is  always  an  exact  measure  of  a  number. 
What  numbers  are  exact  measures  of  21  ?  of  30  ?  of  16  ? 
of  12  ?  of  48  ?     Is  24  an  exact  measure  of  48  ? 


88  REVIEW  OF  INTEGERS  AND  DECIMALS 

10.  All  the  prime  factors  of  a  number  may  be  fovind 
by  dividing  the  number  by  one  of  its  prime  factors,  and 
dividing  each  quotient  in  turn  by  one  of  its  prime  factors 
until  the  quotient  is  a  prime  number.  The  prime  factors 
of  60  may  be  found  by  dividing  60  by  2,  and  dividing  the 
quotient  (30)  by  2,  and  this  quotient  (15)  by  3.  The 
last  quotient  (5)  is  a  prime  number.  The  prime  factors 
of  60  are  the  divisors,  2,  2,  3,  and  the  last  quotient,  6. 

11.  Find  the  prime  factors  of  48  ;  of  72  ;  of  80. 

12.  Which  of  the  following  are  prime  numbers  :  27,  17, 
39,  51,  29,  91,  53,  89,  77,  57  ? 

13.  Count  by  5's  to  35,  beginning  with  5.  Numbers 
ending  in  5  and  in  0  are  exactly  divisible  by  5. 

14.  What  numbers  are  exactly  divisible  by  10  ?  by  2  ? 
by  5? 

15.  Write  a  number  the  sum  of  whose  digits  is  3.  Is 
the  number  exactly  divisible  by  3?  Is  51  exactly  di- 
visible by  3  ? 

16.  Write  a  number  the  sum  of  whose  digits  is  6,  9, 12, 
or  some  other  multiple  of  3.  Is  the  number  exactly  di- 
visible by  3  ?  Show  by  several  illustrations  that  the  fol- 
lowing statement  is  correct : 

A  number  is  exactly  divisible  by  S  if  the  sum  of  its  digits 
is  exactly  divisible  by  3. 

17.  Which  of  the  following  are  exactly  divisible  by  3  : 
54,  177,  81,  52,  819,  57,  69,  71,  213,  105,  86,  1612  ? 

18.  Write  a  number  the  sum  of  whose  digits  is  divisible 
by  9. 

A  number  is  exactly  divisible  by  9  if  the  sum  of  its  digits 
is  exactly  divisible  by  9. 

19.  Which  of  the  following  are  exactly  divisible  by  9  : 
54,  504,  522,  711,  827,  218,  745,  891,  5375,  457? 


DIVISIBILITY  89 

20.  "Which  of  the  following  are  exactly  divisible  by  2  ? 
by  3  ?  by  5  ?  by  9  ?  by  10  ?  45,  61,  360,  207,  783,  53, 
540,  117,  102,  107,  37,  97,  201,  855,  732,  380,  4320,  105  ? 

21.  A  number  that  is  exactly  divisible  by  9  is  exactly 
divisible  by  3.     Why  ? 

22.  All  multiples  of  even  numbers  are  even  numbers. 
Why? 

23.  The  number  denoted  by  the  two  right-hand  figures 
of  216  is  16.     Will  4  exactly  divide  16  ?  200  ?  216  ? 

24.  Write  a  number  of  three  or  more  places  in  which 
the  number  denoted  by  the  two  right-hand  figures  is  some 
multiple  of  4.     Is  the   number  exactly  divisible  by  4? 

A  numher  is  exactly  divisible  hy  4  if  the  number  denoted 
by  its  two  right-hand  figures  is  exactly  divisible  by  4. 

25.  Which  of  the  following  are  exactly  divisible  by  4  : 
112,  202,  420,  532,  514  ? 

26.  Centennial  years  that  are  divisible  by  400  (1200, 
1600,  etc.)  and  other  years  divisible  by  4  are  leap  years. 
Which  of  the  following  will  be  leap  years:  1910,  1912, 
1908,  1926,  1924,  1960,  1990,  2000,  2100  ? 

27.  Write  10  numbers  that  are  exactly  divisible  by  3. 

113.  The  three  pairs  of  factors  of  24  are  2  and  12,  3 
and  8,  4  and  6.  Name  all  the  pairs  of  factors  of  each  of 
the  following,  naming  no  factor  larger  than  20  : 


4 

15 

25 

34 

44 

56 

77 

100 

6 

16 

26 

35 

45 

57 

80 

108 

8 

18 

27 

36 

48 

60 

81 

110 

9 

20 

28 

38 

49 

64 

84 

120 

10 

21 

30 

39 

50 

m 

90 

121 

12 

22 

32 

40 

51 

70 

96 

132 

14 

24 

83 

42 

54 

72 

99 

144 

PART   II 

FRACTIONS 

114,   1.    Draw  a  line  4  inches  long.      Divide  it  into 
two  equal  parts.     What  is  each  part  called  ? 

2.  If  6  pupils  are  separated  into  three  equal  groups, 
what  part  of  the  pupils  will  each  group  contain? 

3.  Draw  a  line.      Divide  it   into   eight   equal   parts. 
What  is  each  part  called  ? 


i 

1 

2 

*■ 

i 

i 

■k 

i  ,  i 

i 

^ 

1 
8 

i 

i 

i  ^ 

1     l^\   1     1 

16  \ie\/6\  16 

/il/6 

76  \  76 

1        1 
76  \76 

76  1  76 

/        / 
16  1  /6 

T6\l6 

B 


4.  A  represents  the  line  undivided.  B  represents 
the  line  as  divided  into  two  equal  parts.  What  does 
C  represent?  i>?  J57?  What  name  is  given  to  each  part 
of  the  line  in  jB?  in  (7?  in  i>?  in  ^?  In  each  case,  how 
many  of  the  parts  does  it  take  to  equal  the  entire  line? 

5.  The  length  of  the  line  is  represented  in  turn  by 
1?  f ^  \'>  f  1  and  \\.     1  of  the  line  =  |  =  |  =  ^^g  of  the  line. 

6.  I  of  the  line  =  f  =  ^^^  of  the  line. 

7.  \  of  the  line  +  \  of  the  line  =  f  of  the  line.  \  of 
the  line  —  J  of  the  line  =  f  of  the  line. 

8.  What  is  the  sum  of  \  of  the  line  and  \  of  the  line? 
What  is  the  difference  between  J  of  the  line  and  J  of  the 
line?  \  of  the  line  and  |  of  the  line? 

00 


FRACTIONS  91 

9.  I  of  the  line  is  longer  than  f  of  the  line.  -^^  of  the 
line  is  longer  than  |  of  the  line,  f  of  the  line  is  longer 
than  I  of  the  line. 

10.  Using  8  objects,  show  that  |^  of  8  objects  is  the 
same  as  |  of  8  objects,  and  that  |  of  8  objects  is  the  same 
as  "I  of  8  objects. 

34.  Show  by  dividing  circles  that  |  of  a  circle  is  equal 
to  I  of  a  circle  ;  that  |  of  a  circle  is  equal  to  |  of  a  circle ; 
that  J  of  a  circle  plus  |^  of  a  circle  is  equal  to  |-  of  a 
circle ;  that  f  of  a  circle  is  equal  to  ^|  of  a  circle. 

12.  Show  by  dividing  rectangles  that  |^,  |,  -|,  -j^,  and 
-^  of  a  rectangle  are  equivalent  parts. 

13.  Using  objects,  show  that  ^  of  12  objects  is  the  same 
as  I  of  12  objects  ;  that  J  of  12  objects  is  the  same  as  | 
of  12  objects. 

14.  Show  by  folding  paper  that  i  =  |==  |  =«  ^g- ;  that  J 

=-  1  ^  3  _  _4_ 
6         9  ~  12* 

116.  Ratio. 

A 

S . 

c . 

1.  Line  A  is  what  part  of  line  B  ?  what  part  of  line  0? 

2.  If  B  is  called  1,  what  isA?0?  If  (7 is  called  1, 
what  is  ^  ?  ^  ?     If  (7  is  called  6,  what  is  ^  ?  J5  ? 

3.  If  A  is  called  3,  what  is  ^?  (7?  If  J.  is  called  J, 
what  is  J5?  0? 

4.  The  ratio  of  line  A  to  line  B  is  ^,  What  is  the 
ratio  of  line  A  to  line  (7?  of  line  B  to  line  ^  ?  of  line  B 
to  line  a?  of  line  O  to  line  B  ?  of  line  O  to  line  A  ? 


FRACTIONS 


n 


R 


116.    1.    The  surface  A  is  what  part  of  the  surface 
Bl  oiC^  oiD?  oiU? 

2.  The  ratio  of  ^  to  ^  is ;  of  A  to  (7  is ;  of 

-A  to  2>  is ;  oi  A  to  U  is . 

3.  The  surface  B  is  what  part  of  the  surface  C?  of  D  ? 

of  ^?     The  ratio  of  B  to  (7  is ;  of  J5  to  i)  is ; 

of  5  to  ^  is . 

4.  What  is  the  ratio  of  C  to  U?  If  (7  represents  40  A, 
what  does  U  represent  ? 

5.  What  is  the  ratio  of  ^  to  ^  ?  oi  Oto  A?  of  i>  to 
A7  of  ^  to  A?  of  0  toB?  oiEto  (7?  of  i>to^?  of  ^ 
to  5? 

6.  The  ratio  of  (7  to  D  is  |,  or  §  ;  oi  D  to  C  is  f,  or  |. 

7.  What  is  the  ratio  oiDtoEl  of  ^ to  i>?  oiUtoC? 

8.  If  A  represents  10  acres,  what  does  B  represent  ? 
C?  B?  B? 

9.  If  B  represents  40  acres,  what  does  A  represent  ? 
C?  B?  JEJ? 

10.  If  the  cost  of  the  land  represented  by  B  is  f  100, 
what  is  the  cost  of  the  land  represented  hy  A?  C?  B?  B? 

11.  If  the  area  represented  by  B  is  640  acres,  what  is 
the  area  represented  hy  0?  B?  A?  B? 

12.  Draw  two  lines  such  that  the  ratio  of  one  to  the 
other  is  -J;  |;  2;  5. 

13.  Draw  oblongs  such  that  the  ratio  of  one  to  the  other 


FRACTIONS  93 

117.  1.  The  unit  of  3  is  1  ;  of  3  da.  is  1  da.;  of  3  mi. 
is  1  mi. 

2.  The  unit  1  mi.  may  be  regarded  as  composed  of 
equal  parts,  as  of  2  half  miles,  of  4  quarter  miles,  of  8 
eighth  miles,  etc.  If  the  unit  1  mi.  is  regarded  as  com- 
posed of  4  equal  parts,  each  part  is  expressed  as  ^  mi.  ;  3 
such  parts  are  expressed  as  |  mi.  A  unit  may  be  re- 
garded as  composed  of  2  or  more  equal  parts. 

3.  A  fraction  is  one  or  more  of  the  equal  parts  of  a 
unit,  as  |,  |,  etc. 

4.  In  the  fraction  |,  4  is  the  denominator.  It  shows 
the  number  of  equal  parts  into  which  the  unit  has  been 
divided.  It  names  the  equal  parts.  3  is  the  numerator. 
It  shows  the  number  of  the  equal  parts  of  the  unit  that 
have  been  taken  to  make  the  fraction  |.  |  denotes  3  of 
the  4  equal  parts  of  the  unit  1. 

5.  When  a  unit  is  divided  into  two  or  more  equal 
parts,  each  of  these  parts  becomes  in  turn  a  unit.  Such 
a  unit  is  called  a  fractional  unit.  \,  \,  ^,  etc.,  are  frac- 
tional units.  The  unit  of  ^  is  \,  What  is  the  unit  of 
each  of  the  following  :  |,  |,  |,  |-  yd.,  -f^  yr.  ? 

6.  Draw  a  line  1  ft.  long.  Divide  it  into  4  equal  parts. 
Show  the  part  that  is  expressed  by  \  ft. ;  by  |  ft. ;  by 
^  ft.  The  ratio  of  1  part  of  the  line  to  the  whole  line  is  J. 
What  is  the  ratio  of  2  parts  of  the  line  to  the  whole  line  ? 
of  3  parts  ?  What  is  the  ratio  of  the  line  to  1  part  ?  to  2 
parts  ?  to  3  parts  ? 

7.  Draw  a  line  8  in.  long.  Let  it  represent  1  mi. 
Show  the  part  that  represents  f  mi.;  |  mi.;  |  mi.  Show 
the  part  whose  ratio  to  the  whole  line  is  \^  ■^,  |^, 
|,  |.  Show  the  part  to  which  the  ratio  of  the  whole 
line  is  2;  8;  4;  j;  j;  f 


94  FRACTIONS 

118.  1.  f  f  wk.,  f  yd.,  f  gal.,  IJ  yr.,  |,  J;^  lb.,  f, 
I  mi.,  -Jj^,  3%. 

«.    Read  aloud  each  of  the  above  fractions. 

h.  Tell  into  how  many  parts  the  unit  in  each  has  been 
divided. 

c.  Name  the  unit  in  which  each  is  expressed. 

d.  Tell  how  many  of  these  parts  are  expressed  in  each 
fraction. 

e.  Read  the  denominator  of  each  fraction. 
/.    Read  the  numerator  of  each  fraction. 

g.  Draw  a  line  to  represent  the  unit.  Mark  on  this 
line  the  parts  expressed  in  each  fraction. 

2.  The  numerator  and  denominator  are  called  the 
terms  of  the  fraction. 

3.  A  fraction  whose  numerator  is  less  than  the  denomi- 
nator is  called  a  proper  fraction,  as  |,  ^,  etc.  Name  ten 
proper  fractions. 

4.  A  fraction  whose  numerator  is  equal  to  or  greater 
than  the  denominator  is  called  an  improper  fraction,  as 
J,  |,  etc.     Name  ten  improper  fractions. 

5.  When  a  number  is  composed  of  an  integer  and  a 
fraction,  it  is  called  a  mixed  number.  Q\  is  a  mixed  num- 
ber. Its  value  is  expressed  in  two  different  units.  The 
6  is  expressed  in  units  of  ones;  the  -J  is  expressed  in  units 
of  one  jifths.     Name  ten  mixed  numbers. 

6.  The  value  of  1,  expressed  in  the  fractional  unit  ^, 
is  f ;  of  2  is  ^ ;  of  3  is  f ;  of  4  is  f  ;  of  5  is  f ;  of  8  is  f. 

7.  What  kind  of  a  number  is  5|?  In  what  unit  is  5 
expressed?  In  what  unit  is  |  expressed?  The  value  of 
6|  may  be  expressed  in  the  fractional  unit  J.  There  are 
I  in  1.  In  5  there  are  5  times  |,  or  ^,  ^  and  |  are  ^. 
What  kind  of  a  fraction  is  ^? 


REDUCTION  96 

REDUCTION 

119.  Changing  Mixed  Numbers  to  Improper  Fractions. 

Change  4|-  to  an  improper  fraction. 

Model  :  5  times  4  is  20 ;  20  and  3  are  23  ;  write  23  over  the  de- 
nominator, thus :  \^. 

To  change  a  mixed  number  to  an  improper  fraction^  mul- 
tiply the  integer  hy  the  denominator  of  the  fraction^  add  the 
7iumerator,  and  write  the  sum  over  the  denominator  of  the 
fraction. 

120.  Oral  Exercises. 

Change  the  following  to  improper  fractions :  * 


a 

b       c 

d 

e 

/ 

ff 

^              *             J 

ifc 

1. 

H 

n  8f 

H 

5| 

4f 

H 

2|     1|    8i 

8f 

2. 

9| 

5|    7f 

^ 

n 

9* 

H 

2|    H    H 

7i 

3. 

5f. 

5i    If 

^ 

2f 

8* 

H 

7f     4|    6f 

5f 

4. 

2f 

9f    7^ 

41 

n 

5i 

6f 

3f     8f    2f 

9^ 

5. 

2| 

9f    7f 

6t 

H 

7| 

6f 

4f     2A    7f 

s* 

6. 

6* 

2*31 

^ 

n 

3i'j 

9t% 

7t^  5^  8i«J 

6* 

7. 

4i^ 

8fi  9i\ 

3f 

7| 

6t\ 

7i\ 

5A  7A  4^ 

5A 

8.  Write  ten  mixed  numbers  and  change  them  to  im- 
proper fractions. 

9.  Express  the  value  of  the  following  integers  in  the 
fractional  unit  ^:  3,  5,  7,  6,  9,  2,  8,  10,  12. 

10.    Write  ten  proper  fractions.     State  what  the  frac- 
tional unit  is  in  each. 

U.    Change  to  improper  fractions :  3|  yd.,  4|  in.,  8|-  mi. 

*  This  exercise  contains  practically  all  the  combinations  in  addition 
and  multiplication.    It  should  be  used  frequently  as  a  review  exercise. 


96  FRACTIONS 

121.  Changing  Improper  Fractions  to  Whole  or  Mixed 
Numbers. 

1.  What  kind  of  a  fraction  is  ^?  What  is  the  unit 
in  which  its  value  is  expressed?  How  many  of  these  frac- 
tional units  does  it  take  to  make  the  unit  1  ?  How  many 
units  of  1  are  there  in  ^?  in  ^?  in  J^?  in  -1^3.?  in  J^? 

2.  What  does  the  denominator  of  a  fraction  show? 
Which  term  of  the  fraction  tells  the  number  of  the  frac- 
tional units  it  takes  to  make  a  unit? 

To  change  an  improper  fraction  to  a  whole  or  a  mixed 
number,  divide  the  numerator  hy  the  denominator, 

122.  Oral  Exercises. 

Change  the  following  to  whole  or  mixed  numbers :  * 


a 

b 

c 

d 

e 

/ 

9 

A 

i 

y 

A 

1. 

¥ 

V- 

V 

Y 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

2. 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

3. 

¥ 

¥ 

h^ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

4. 

V 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

5. 

Y 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

¥ 

6. 

¥ 

\l 

¥ 

¥ 

¥ 

M 

¥^ 

\l 

^ 

¥i^ 

If 

7. 

M 

W 

¥i^ 

¥ 

¥ 

a 

!! 

\\ 

!f 

!i 

\{ 

8.  Write  ten  improper  fractions  and  change  them  to 
whole  or  mixed  numbers. 

9.  Write  ten  mixed  numbers  and  change  them  to  im- 
proper fractions. 

♦  This  exercise  contains  nearly  all  of  the  facts  of  division  and  subtrac- 
tion.   It  should  be  used  frequently  as  a  review  exercise. 


ADDITION  AND  SUBTRACTION  97 

ADDITION  AND  SUBTRACTION  OF  FRACTIONS 
123.  Oral  Exercises. 

1.  What  is  the  sum  of  2  books  and  3  books  and  1 
book  ?  Are  these  quantities  expressed  in  the  same  unit 
of  measure  ? 

2.  Name  three  quantities  that  are  not  expressed  in  the 
same  unit  of  measure.     Can  their  sum  be  found  ? 

3.  Are  the  following  fractions  expressed  in  the  same 
unit  of  measure :  fi  f ?  i  ?  Fractions  that  are  expressed 
in  the  same  unit  of  measure  are  said  to  be  similar 
fractions.  Similar  fractions  have  the  same  denominator. 
Only  fractions  that  are  expressed  in  the  same  unit  of 
measure  can  be  added. 

4.  The  sum  of  ^,  |,  |,  and  |  is  |,  which  is  equal  to  2^. 

5.  Add  2|  ft.,  5^  ft.,  and  6J  ft. 


Modi 

!l:       ^ 

ft. 

Add  the  fractions  first 

:  J  ft.  and  J  ft.  and  1  ft. 

5J 

ft. 

are  %  ft., 

which 

are  equa! 

1  toli 

ft.     Write  }  ft. 

6i 
14^ 

ft. 

below  the  column  of  fractions, 

and  carry : 

Lft.to 

ft. 

the  column  of  whole  numbers. 

6. 

Add  the  following : 

a 

h 

G 

d 

e 

/ 

ff 

h 

2 

6i 

3 

H 

4f 

H 

H 

6| 

7^ 

5t% 

8 

7* 

9f 

9f 

U 

6 

n 

5 

6if 

3f 

^ 

4 

^ 

H 

H 

H 

6^ 

8t% 

H 

4* 

7i 

2 

H 

4f 

^ 

4t^ 

7x^ 

5i 

n 

8* 

8+ 

6i 

2i 

^ 

3^ 

3t^ 

4| 

6| 

5i 

1\ 

5J 

7f 

9| 

8iV 

m 

8 

2* 

6| 

^ 

»i 

n 

6| 

7* 

9^ 

98 


FRACTIONS 


124:,  Oral  Exercises. 


1.  }  ft.  -  J  ft.  =  f  ft.      I  mi.  -  I  mi.  =  f 
-A  yj*-  =  1%  y^'     f  da.  -I  da.  =  f  da. 

2.  6f  ft. -4  ft.  = ft.     i8|-$5  =  $- 


mi. 


Hyr- 


mi. 


2mi. - 

mi. 

3.    Subtract 

>  the 

fractions    first 

and  then 

the   whole 

numbers : 

•• 

a 

8f  wk. 
If  wk. 

h 

HI 

s/yd. 

3|  yd. 

d 

^  yr. 

6|lb. 
4flb. 

/ 

8f  wk. 
5    wk. 

3-A 

in. 
in. 

h 

t9| 

24|  yd. 
17i  yd. 

29|  yd. 
19|  yd. 

4.  Find,  the  sum  of  each  of  the  above. 

5.  Subtract  3J  ft.  from  6  ft. 
Since  there  is  no  fractional  part  in  the  minu- 


Model  :      6    ft. 

end,  the 

suna  of  the  fraction  of  the  subtrahend 

SI  ft. 

and  the  fraction  of  the  difference  is  : 

I  ft.  i  ft. 

2|ft. 

and  I  ft. 

are  1  ft. 

Carry  1 

ft.  to  3  ft.  and  sub- 

tract  tht 

)  integers. 

4  ft.  and  2  ft.  are  ( 

3  ft. 

6.    Subtract : 

8    hr.       1    da. 

19 

16 

5     ft. 

8    lb. 

9    mi. 

4|  hr.       2J  da. 

$H 

$2| 

If  ft. 

311b. 

6|mi. 

7.    Subtract: 

7fyd.     gjyr. 

8   A. 

6|mi. 

5f  wk. 

6   yd. 

8|hr. 

2Jvd.      6    yr. 

4J  A. 

3|mi. 

3f  wk. 

3fvd. 

4    hr. 

8.    If  a  boy  attended  school  3|  da.  in  a  certain  week, 
how  many  days  was  he  absent  ? 


ADDITION  AND  SUBTKACTION  99 

9.  A  girl  who  was  taking  lessons  on  the  piano  prac- 
ticed as  follows  during  one  week :  Monday,  1 J  hr. ;  Tuesday 
morning,  1  hr. ;  Tuesday  afternoon,  |  hr. ;  Wednesday, 
Ifhr.;  Thursday,  I  hr. ;  Friday,  1^  hr. ;  Saturday  morn- 
ing, If  hr. ;  Saturday  afternoon,  |  hr.  How  many  hours 
did  she  practice  during  the  week  ? 

10.  A  boy  had  10  mi.  to  travel.  If  he  traveled  3| 
mi.  on  foot  and  rode  the  remainder  of  the  distance,  how 
far  did  he  ride  ? 

125.  Oral  Exercises. 

1.  Subtract  6|  from  9f . 

Model  :       91  The  sum  of  |  and  the  fraction  of  the  difference 

p4      is  1§.     Find  what  must  be  added  to  f  to  make  1  and 

93  add  it  to  I .  f  and  ^  are  1.  |  and  f  are  f.  Carry  1 
to  6.  7  and  2  are  9. 
The  nnmerator  of  the  fraction  in  the  difference  may  be  found  by 
subtracting  4  (the  numerator  of  the  fraction  in  the  subtrahend)  from 
5  (the  denominator  of  the  fraction  in  the  minuend^,  and  adding  2 
(the  numerator  of  the  fraction  in  the  minuend).  Explain  why  this 
method  will  give  the  correct  result.     Use  this  method*  in  subtracting. 

2.  Subtract  without  the  use  of  a  pencil : 


a 

b 

c 

d 

e 

/ 

ff 

h 

I 

6| 

n 

n. 

H 

9H 

Vj 

m 

6if 

n 

21 

6f 

3i 

H 

4if 

sji 

m 

ii! 

^ 

6t^ 

H 

n 

m 

^ 

8i^ 

9t^ 

6t^ 

m 

?i 

M 

M 

m 

H 

2H 

14 

4^^ 

m 

3.  Find  the  sum  of  each  of  the  above  exercises. 

4.  A  dressmaker  had  two  pieces  of  cloth  containing 
8|  yd.  and  6|  yd.,  respectively.  She  used  lOJ  yd.  in 
making  a  dress.     How  much  cloth  was  left? 


100 


FRACTIONS 


REDUCTION 
126.   Changing  to  Higher  and  Lower  Terms. 


w. 

« 

M 

» 

s 

^ 

^ 


1.  If  A  represents  a  unit  divided  into  2  equal  parts, 
what  does  ^  represent  ?   C?  i>?  ^?  J'? 

2.  The  fractional  unit  of  ^  is  \\  what  is  the  fractional 
unit  oi  CI  Bl  El  Fl  What  part  of  the  fractional  unit 
of  A  is  the  fractional  unit  of  ^?  (7?  i>?  ^?  ^?  How 
many  of  the  fractional  units  of  B  does  it  take  to  make  one 
of  the  fractional  units  of  -4.  ?  How  many  of  (7?  of  2>? 
of  El  of  Fl 

3.  The  fractional  unit  \  is  what  part  of  the  fractional 
unit  J  ?  I  =  |.     I  is  what  part  of  J  ? 

4.  The  denominator  of  the  fractional  unit  ^  is  2  times 
the  denominator  of  the  fractional  unit  \.  It  shows  that 
the  unit  has  been  divided  into  twice  as  many  equal  parts. 
It  will  therefore  take  2  of  the  fractional  units  sixteenths  to 
make  one  of  the  fractional  units  eighths.     i%=|- 

5.  The  fractions  J^  f  ^  f  ^  ^  are  the  same  in  value.  They 
differ  in  form.  Changing  the  form  of  a  fraction  without 
changing  its  value  is  called  reduction. 

6.  The  fraction  |  is  equal  to  the  fraction  ■^.  Compare 
their  numerators.  8  is times  4.  Compare  their  de- 
nominators.    16  is times  8.     How  may  ^  be  derived 

from  I  ?     How  may  }  be  derived  from  ^  ? 


REDUCTIOlSr-'''  •    "  •         '    '        101 

'  •'•  *    1  •.*  '  ''*»?  *  V, '.  A 

7.  Compare  in  a  similar  way  tlie  tet'nis  of  the'  fractions 
I  and  ^2  5  \  ^"^  1^5  f  ^^^  A*  What  effect  upon  the 
value  of  a  fraction  has  multiplying  both  terms  by  the  same 
number  ? 

8.  A  fraction  is  an  indicated  division.  |  is  the  same 
as  6}^.  The  denominator  of  the  fraction  is  the  divisor,  and 
the  numerator  is  the  dividend.  What  effect  upon  the 
quotient  has  multiplying  both  the  dividend  and  the  divisor 
by  the  same  number  ?  Is  multipl3dng  both  the  numerator 
and  the  denominator  of  a  fraction  by  the  same  number 
the  same  as  multiplying  both  the  dividend  and  the  divi- 
sor by  the  same  number  ?     |  =  |  =  ^^  =  j^. 

9.  The  fraction  \^  is  equal  to  the  fraction  |-.  Compare 
their  numerators.  5  is  what  part  of  10  ?  Compare  their 
denominators.  8  is  what  part  of  16  ?  Compare  in  a  sim- 
ilar way  -5^2  with  |;  -^^  with  |.  What  effect  upon  the 
value  of  a  fraction  has  dividing  both  terms  by  the  same 
number?     3-^  =  |  =  |=f. 

10.  What  effect  upon  the  quotient  has  dividing  both 
dividend  and  divisor  by  the  same  number  ?  Is  dividing 
both  numerator  and  denominator  of  a  fraction  by  the  same 
number  the  same  as  dividing  both  dividend  and  divisor  by 
the  same  number  ? 

Multiplying  or  dividing  both  terms  of  a  fraction  hy  the 
same  number  does  not  alter  the  value  of  the  fraction. 

11.  Change  the  form  of  the  following  without  changing 
their  value :    f ,  ^,  f ,  if,  ^,  If,  |^,  ^. 

12.  By  what  must  the  terms  of  the  fraction  ^  be  multi- 
plied to  reduce  the  fraction  to  lOths  ?  to  15ths  ?  to  20ths  ? 

13.  How  many  12ths  are  there  in  1  ?  in  ^  ?  in  |  ? 
in|^ 


102  FRACTIONS 

127.    Written  Exercises. 

1.  Change  |  to  12ths.  As  the  denominator  12  is  4 
times  the  denominator  of  |,  the  numerator  of  the  required 
fraction  must  be  4  times  the  numerator  of  |. 

Model:     ^^f^-     3   is  contained  in  12  four  times. 
4  times  2  is  8.     |  =  -j^. 

Another  method.      1  =  H  i  i  =  i  of  [f ,  or  ^^2 ;  |  =  2  tunes  ^^^  or  ^. 

2.  State  how  you  would  find  the  number  that  3  must 
be  multiplied  by  to  change  |  to  20ths. 


3.  Change  to  12ths 

4.  Change  to  18ths 

5.  Change  to  24ths 

6.  Change  to  20ths 

7.  Change  to  36ths 

8.  Change  to  30ths 


2'  3'  t'  6'  3'  I'  h  6'  f  • 
2'  3'  6'  9'  6'  9'  6'  9'  "S* 
f'  I'  '2'  6'   8'    1^'  t'  A* 

2'  I'  h  A'  I'  10'  h  tV- 

I'  h  i  I  h  {h  tV  h  i' 

iV  ^u^  h  h  h  h  \h  h  A- 
9.    Write  eight  fractions  and  change  them  to  48ths. 

10.  When  the  terms  of  a  fraction  have  been  made  larger 
by  reduction,  the  fraction  is  said  to  have  been  reduced  to 
higher  terms. 

128.   Oral  Exercises. 

1.  Express  each  in  a  different  form  without  changing 
the  value:  f,  8|,  ^  J,  2,  2^,  |,  f,  7^,  J^,  |f,  1. 

2.  Find  the  sum  of  3J,  2|,  and  7. 

3.  Find  the  difference:  gf^?!    |?|    S| 

4.  Show  that  J  of  2  yd.  is  equal  to  |  of  1  yd.;  that 
J  of  8  ft.  is  the  same  as  |  of  1  ft. 


REDUCTION  103 

129.  Oral  Exercises. 

1.  Show  by  a  diagram  that  f  of  a  line  is  equivalent  to 
f  of  the  line  ;  that  J  f t.  =  f  ft.  ;  that  |  ft.  +  J  ft.  =  |  ft., 
or  11  ft. 

2.  Show  with  objects  that  |  of  12  objects  =  3^2  of 
12  objects;  that  1  of  12  objects  =  ^^2  ^^  ^^  objects;  that 
f  of  12  objects  +  i  of  12  objects  =  -^^  of  12  objects  +  -^^  of 
12  objects,  or  \^  of  12  objects. 

3.  Why  is  it  necessary  to  change  |  and  |  to  12ths  be- 
fore finding  their  sum  ? 

4.  Show  by  a  diagram  that  the  difference  between  J  ft. 
and  \  ft.  is  ^^2  ^^' 

5.  If  I  of  a  group  of  objects  contains  6  objects,  show 
the  number  of  objects  in  |  of  the  group. 

6.  Show  with  objects  that  if  |  of  a  group  of  objects  is 
6  objects,  the  whole  group  contains  9  objects. 

7.  Show  by  a  diagram  that  if  |  of  the  length  of  a  line 
is  4  ft.,  the  entire  length  of  the  line  is  10  ft. 

8.  If  6  objects  represent  |  of  the  number  of  books  on 
a  certain  shelf,  represent  by  objects  ^  of  the  number  of 
books  on  the  shelf.     Represent  all  the  books. 

9.  What  is  the  least  number  of  boys  that  may  be 
separated  either  into  groups  of  3  boys  or  of  4  boys  ? 

10.  What  is  the  least  number  of  equal  parts  into  which 
a  rectangle  can  be  divided  so  that  either  ^  or  -J  of  the 
rectangle  may  be  shown  ? 

11.  What  is  the  least  number  of  girls  that  can  be  sepa- 
rated into  groups  containing  as  many  girls  as  are  indicated 
in  the  denominator  of  any  one  of  the  following  :  J,  f ,  f ,  f  ? 


104  FRACTIONS 

130.  Common  Factors. 

1.  The  exact  measures  of  12  ft.  are  1  ft.,  2  ft.,  3  ft.,  4 
ft.,  and  6  ft.  Name  the  exact  measures  of  18  ft.  Which 
are  exact  measures  of  both  12  ft.  and  18  ft.?  1  ft.,  2  ft., 
3  ft.,  and  6  ft.  are  common  measures  of  12  ft.  and  18  ft. 

2.  A  number  that  is  a  factor  of  two  or  more  numbers  is 
called  a  common  factor,  or  a  common  measure,  of  the  numbers. 

3.  Name  the  common  factors  of  18  and  24.  Name  the 
largest  common  factor  of  18  and  24. 

4.  The  greatest  common  factor  of  two  or  more  numbers 
is  the  largest  number  that  will  exactly  divide  each  of 
them.  This  is  also  called  the  greatest  common  measure^  or 
the  greatest  common  divisor^  of  the  numbers. 

5.  Draw  a  line  12  in.  long  and  another  line  18  in.  long. 
What  is  the  longest  measure  that  can  be  applied  without 
a  remainder  in  measuring  both  of  these  lines  ?  What  is 
the  greatest  common  measure  of  12  ft.  and  18  ft.? 

6.  Show  with  objects  or  by  a  diagram  all  the  common 
measures  of  8  objects  and  12  objects. 

131.  Name  the  exact  measures  of  : 


1. 

10  ft. 

6. 

30  mi. 

11. 

50  rd. 

16. 

$80 

2. 

16  gal. 

7. 

36  yd. 

12. 

27  pt. 

17. 

190 

3. 

20  da. 

8. 

40  yr. 

13. 

28  da. 

18. 

$60 

4. 

18  hr. 

9. 

481b. 

14. 

64  ft. 

19. 

175 

5. 

24  in. 

10. 

45  qt. 

15. 

72  mi. 

20. 

$84 

132.   Name  the  greatest  common  factor  of  : 

1.  6  and    8     4.  12  and  18     7.  12  and  48     10.  20  and  60 

2.  8  and  12     5.  24  and  30     8.  15  and  30     11.  18  and  36 

3.  9  and  12     6.  24  and  36     9.  30  and  36     12.  14  and  28 


REDUCTION  105 

133.   Oral  Exercises. 

1.  The  fractional  unit  J  must  be  repeated  how  many 
times  to  equal  the  fractional  unit  ^  ?  The  fractional  unit 
^^  must  be  repeated  how  many  times  to  equal  the  frac- 
tional unit  -^^  ? 

2.  By  what  must  the  terms  of  the  fraction  |^|  be  divided 
to  give  the  fraction  ^2  •   i  M  M  ^  2  ^ 

3.  When  the  terms  of  a  fraction  have  been  made  smaller 
by  reduction,  the  fraction  is  said  to  have  been  reduced  to 
lower  terms.  A  fraction  is  in  its  lowest  terms  when  the 
terms  have  no  common  factor. 

4.  The  fraction  \^  is  not  in  its  lowest  terms,  as  both 
terms  are  exactly  divisible  by  5.  Which  of  the  following 
are  in  their  lowest  terms  :  |,  f,  |,  |,  1|,  if,  IJ.  |f  ? 

5.  Dividing  both  terms  of  a  fraction  by  a  common  fac- 
tor is  called  canceling  the  common  factor. 

In  reducing  a  fraction  to  its  lowest  terms,  cancel  in  turn  the  largest 
factors  that  are  seen  to  be  common  to  both  teims.  Canceling  the 
greatest  common  factor  of  both  terms  reduces  the  fraction  to  its  low- 
est terms . 


134.   Oral  Exercises. 

Reduce  to  lowest  terms  ; 

1. 

1^2'  tI'  iV  ^2'  A'  it 

9. 

II.  If.  If.  If.  M.  11 

2. 

A'  A'  ^¥^  A'  A'  A 

10. 

A.  ^.  If.  M.  1 2'  H 

3. 

M^if^Maili^il 

11. 

If.  f  I.  ^i.  M.  If.  ft 

4. 

/o^  A'  -io'  A'  ro'  A 

12. 

fi.fMi.if.ii.H 

5. 

il  ih  il'  ih  ih  u 

13. 

if.  1!.  ih  If.  If.  If 

6. 

ih  /r  B^6^  A'  ih  Jl 

14. 

if.  Ih  ih  f«.  If.  iV\ 

7. 

U^  M^  1%  iV  A'  A 

15. 

it  1%.  If.  li.  II.  H 

8-  M.  H.  if.  if.  H-  It        16.  if,  if  if  H,  T%,  ,vk 


106  FRACTIONS 

135.  Multiples. 

1.  2  is  a  factor  of  4,  6,  8,  10,  etc.  Each  of  these  num- 
bers is  a  multiple  of  2.     Name  the  multiples  of  3  to  27. 

2.  A  number  that  is  exactly  divisible  by  a  given  num- 
ber is  called  a  multiple  of  the  given  number.  Name  a  mul- 
tiple of  6 ;  of  8 ;  of  7. 

3.  Write  the  multiples  of  3  to  27  and  of  4  to  36. 
Which  of  the  numbers  written  are  multiples  of  both  3  and 
4  ?  These  numbers  are  common  multiples  of  3  and  4. 
Which  is  the  least  multiple  common  to  3  and  4  ? 

4.  A  number  that  is  a  multiple  of  each  of  two  or  more 
numbers  is  called  a  common  multiple,  and  the  least  number 
that  is  a  common  multiple  of  each  of  two  or  more  numbers 
is  called  the  least  common  multiple  of  the  numbers. 

5.  Write  all  the  multiples  of  4  to  36  and  of  6  to  54. 
Name  the  multiples  common  to  4  and  6.  Which  of  these 
is  the  least  common  multiple  of  4  and  6  ? 

6.  Name  the  least  common  multiple  of  3  and  4 ;  of  2, 

3,  and  4.  Since  4  is  a  multiple  of  2,  the  least  common 
multiple  of  2,  3,  and  4  is  the  same  as  the  least  common 
multiple  of  3  and  4.     Tlie  least  common  multiple  of  2,  3, 

4,  and  6  is  the  same  as  the  least  common  multiple  of  4  and 
6.     Why? 

7.  Write  four  numbers  such  that  the  least  common 
multiple  of  the  numbers  is  the  same  as  the  least  common 
multiple  of  some  two  of  the  numbers. 

8.  In  finding  the  least  common  multiple  of  2,  3,  4,  and 
9,  which  numbers  need  not  be  considered,  and  why  ? 

9.  Find  the  least  common  multiple  of  3,  5,  and  7.  Show 
by  several  illustrations  that  the  least  common  multiple  of 
two  or  more  prime  numbers  is  their  product. 


MULTIPLES  107 

136.   Name  the  least  common  multiple  of : 

1.    4  and  5               7.    8  and  12  13.  10  and  12 

8.  6  and  9  14.  12  and  9 

9.  4  and  10  15.  3  and  7 

10.  5  and  10  16.  2,  3,  and  6 

11.  10  and  4  17.  3,  4,  and  8 

12.  5  and  8  18.  4,  5,  and  15 


2.  4  and  6 

3.  3  and  4 

4.  6  and  8 

5.  5  and  7 

6.  4  and  8 

137.  1.  Find 
and  48. 

Model  :    ^ 

48 

Find  the  least  common  multiple  of  8,  10,  18, 

As  48  is  a  multiple  of  8,  cancel  8.  As  the 
factor  2  is  common  to  10  and  to  48,  cancel  this 
factor  of  10,  leaving  the  factor  5.  As  6  is  a 
factor  common  to  18  and  48,  cancel  this  factor 
of  18,  leaving  the  factor  3.  5  x  3  x  48,  or  720, 
is  the  least  common  multiple  of  8,  10,  18,  and 
48. 

2.  In  finding  the  least  common  multiple  of  3,  4,  6,  9, 
and  12,  which  numbers  may  be  canceled?  From  which 
number  may  a  factor  be  canceled? 

3.  In  finding  the  least  common  multiple  of  4, 12,  7,  and 
35,  which  number  may  be  canceled  because  it  is  a  factor  of 
12  ?    Which  may  be  canceled  because  it  is  a  factor  of  35  ? 

138.    Find  by  inspection  the  least  common  multiple  of: 

7.  10,  15,  25,  40    13.    7,  35,  45,  90,  70 

8.  36,  48,  60,  72    14.    5,  14,  42,  60 

9.  12,  18,  24,  36    15.    6,  7,  8,  9,  84 

10.  3,  5,  30,  45        16.    20,  24,  30,  100 

11.  4,18,27,  72     17.   4,9,20,54 

12.  8,  12,  15,  60      18.    6,  15,  24,  86 


1. 

4,  5,  8,  24 

2. 

3,  12,  15,  30 

3. 

5,  8,  25,  40 

4. 

2,  6,  15,  45 

5. 

7,  21,  49,  84 

6. 

9,  15,  36,  60 

108  FRACTIONS 

ADDITION  AND  SUBTRACTION 
139.   Written  Exercises. 
1.    Change  to  12ths  and  add  |,  |,  f . 
Model  : 


l=* 

2. 

3. 

4. 

5. 

6. 

7. 

l  =  A 

I 
6 

1 

f 

i* 

§ 

A 

S  =  M 

i 

1 

i 

i 

i 

f 

!J  =  2^J=2J 

i 

6* 

1 

i 

8.  Add  the  fractions  in  Exs.  3-8,  Sec.  127. 

9.  Change  to  24ths  and  add  3f,  4 J,  6l|,  7|. 

Model:       3f    =  31f 

41    __  4_4  The  sum  of  the  fractions  is  f  |,  which 

f\ll  _  r2  2  reduces  to  2|.     Write  f  as  the  frac- 

12  ~  ^21  tional  part  of  the  answer.     Carry  2  to 

*t    ~  ^2ir  the  column  of  integers. 
22f 

10.  Change  to  12ths  and  add  4f,  3f,  4f ,  5J,  6^\. 

11.  Change  to  18ths  and  add  8f,  7|,  9J,  6^,  51^. 

12.  Change  to  24ths  and  add  7|,  6^^,  9J,  8|,  4J. 

13.  Change  to  36ths  and  add  311  7i|,  sj,  51  9f . 

14.  Change  to  48ths  and  add  7 J,  9f,  3^,  5||,  8^^. 

15.  Change  to  72ds  and  add  8^,  9f,  7^,  18^^,  3|. 

16.  Reduce  to  lowest  terms:  ||,  ||,  -||,  |-|. 

17.  Change  to  improper  fractions:  7|,  9|,  8|,  7|. 

18.  Change  to  mixed  numbers  :  -^,  -j^,  -^1,  -^. 

19.  Add  4f ,  6f  3f  8f  9f ,  7f 

20.  Change  4  to  12ths  ;  3  to  18ths ;  5  to  20ths. 

21.  Write  ten  fractions  and  reduce  them  to  lower  terms. 


ADDITION  109 

140.  Oral  Exercises. 

1.  What  is  the  least  common  multiple  of  2,  3,  and  4  ? 
of  3,  5,  and  6  ?  of  4,  5,  and  6  ?  of  4,  6,  and  8  ?  of  3,  6,  and 
9  ?  of  5,  8,  and  12  ? 

2.  Can  you  add  the  following  fractions  without  first  re- 
ducing them  :  |^,  |^,  |-?  Are  they  expressed  in  the  same 
fractional  unit  ? 

3.  Can  you  add  the  following  fractions  without  first  re- 
ducing them  :  |,  |,  and  J  ?  Are  they  expressed  in  the 
same  fractional  unit  ?  Only  fractions  that  are  expressed 
in  the  same  fractional  unit  can  be  added. 

4.  What  is  the  unit  of  measure  in  f  ft.  ?  |  ft.  ?  -^  ft.  ? 
These  fractions  may  be  expressed  in  the  same  unit  of 
measure,  -^  ft.    |  ft.  =  f^  ft.    f  ft.  =  f^  ft. 

5.  Can  the  following  be  expressed  in  the  same  unit  of 
measure  :  -J  ft.,  |  da.,  and  ^  gal.?  Can  the  following  : 
1  ft.,  f  ft.,  and  iV  ft.? 

6.  What  is  the  least  common  multiple  of  the  denomi- 
nators of  the  fractions  |^,  f ,  |,  and  ^  ?  The  least  common 
multiple  of  the  denominators  of  two  or  more  fractions  is 
called  their  least  common  denominator. 

141.  Reduce  the  fractions  to  fractions  having  the  least 
common  denominator,  and  add  ; 


a 

h 

c 

d 

e 

/ 

9 

h 

i 

H 

n 

H 

3A 

4iJ 

3^ 

5! 

^ii 

6f 

6f 

H 

6f 

4| 

n 

8| 

8A 

m 

8f 

n 

^ 

7^1 

n 

6f 

6-11 

^ 

n 

4i^ 

4A 

6il 

8i 

H 

n 

^ 

5i 

H 

6« 

n 

n 

5i 

7f 

5A 

6J 

n 

8f 

n 

110  FRACTIONS 


u 

2.  Written  Exercises.* 

1. 

From  5f  subtract  a§. 

Model  : 

5|  = 
8|  = 

5-5^          Reduce 
3^      the  least 
2^1^      tract. 

Subtract 

: 

a 

b 

0 

i 

2. 

6* 

^ 

8| 

5f 

6f 

7f 

8fl 

Reduce  the  fractions  to  fractions  having 
the  least  common  denominator,  and  sub- 


e         f  g         h 

2^      9|     23^      43f 


■"""' 

"~~" 

"""■ 

3.       9| 

6| 
4| 

8i 
6f 

6i 

li 

5| 
2f 

6| 

fi 

4.     5i 
4f 

6f 
4t^ 

7^ 
2| 

9* 

18? 

7| 

19| 
9| 

12i 
8t\ 

15i 
8| 

B.    21f 
10| 

80| 
9 

20 
9! 

26f 
4| 

43| 

7 

16 

101 

18| 
81 

29| 
9^ 

6.    87^ 
64f 

79^ 

90A 
3TA 

68J, 
19t\ 

74| 

20 

13^f 

20 
9H 

4f 
8£ 

143.   1.    Add  each  exercise  in  Sec.  142. 

2.  The  lengths  of  the  bhicl^boards  in  a  certain  school- 
room are  12J  ft.,  14J  ft.,  8|  ft.,  and  GJ  ft.,  respectively. 
Find  the  combined  length  of  the  four  blackboards. 

3.  Find  the  difference  in  the  weight  of  two  turkeys,  if 
one  of  the  turkeys  weighs  22J  lb.  and  the  other  17|  lb. 

«  See  Sec.  125. 


ADDITION  AND  SUBTKACTION  111 

U4.   Review  Sec.  134. 
Reduce  to  lowest  terms : 

1-  Ih  If  ih  il  Ih  If      5-  If.  t\\'  AV  1%'  m 

2-     100'  TOO"'  TOO'  T0T7'  Too        ®-  T2'  "50'  yS'  ■g'O'   QO"'  80 

3        70      JLO__    _9JL    JL5_    _4  5_       7  100    JTS      _2_5_    _5JL    __4  0_ 

**•      100'   100'   100'   100'    100        ''  150'   150'   150'  200'  200 

*.     1^%'  ^¥t'  1^5'  T^A'  ilt      8.  ^,  ^%'  Hi  if^'  ii 

145.  Review  Sec.  120. 
Reduce  to  improper  fractions : 

1.  14|,  30J,  161,  331  66f     4.  17f  mi.,  51  yd.,  8f  wk. 

2.  16f,  37f ,  111,  9_i_,  28f     5.  231  ft.,  8f  lb.,  9f  wk. 

3.  38^^,  5f,  16f ,  25|,  671     g.  6^^  yd.,  4J  mi.,  811  A. 

146.  Review  Sec.  122. 

Reduce  to  integers  or  mixed  numbers : 

1.  101  iQjQ.,  loii^  i.Qii,  loii     5.  ^  yd.,  ■^<I  gal.,  ^  wk. 

2.  i|ii,  ian,  ij(^(i,  i_o^o.,  j^^    6.  |j^^,  1^8.,  iJ^,  1^,  1^ 

3.  W.  W,  -W'  V.^-'  ¥/     7-  ¥  da.,  ^/  mi.,  ^2.  gal. 
4-    i^'  4^  ¥'  ¥'  ¥'  ¥'     8.  ^^1  T.,  1/  A.,  ^g<l  bu. 


147. 

Add: 

« 

5 

e? 

c? 

e 

/ 

5' 

43f 

64J 

l^ 

7Tf 

m 

39^ 

66| 

691 

58f 

79J 

24| 

67J 

34| 

76J 

73-1 

74| 

83i 

661 

58| 

65i 

84f 

87tV. 

''56J 

95^ 

78* 

49f 

47^ 

59| 

2 

FRACTIONS 

148.  Subtract: 

a            h 
1.  190f   265| 
137|   124f 

c              d 
398f   443^^ 
154f   217| 

178§ 
25| 

/ 

296J 
180| 

9 

467f 
337| 

2.    Add  each  of  the  above  exercises. 

149.    Review  Sec.  137. 

Give  the  least  common  multiple  of : 


1. 

2,3,4 

8. 

4,5,  8 

15. 

T, 

8,  9 

2. 

3,4,5 

9. 

3,4,  7 

16. 

3, 

4,  6,  8,12 

3. 

4,  5,  6 

10. 

6,7,  8 

17. 

5, 

7,  8,12,  4 

4. 

2,3,5 

11. 

3,7,  9 

18. 

3, 

5,  6,  8,  15 

5. 

3,4,7 

12. 

4,7,  9 

19. 

7, 

9,  12,  14,  21 

6. 

4,6,8 

13. 

6,  8,  12 

20. 

8, 

10,  12,  15,  20 

7. 

5,7,8 

14. 

5,  8,  12 

21. 

4, 

6,  10,  14,  20 

150.  When  the  least  common  multiple  of  the  denomi- 
nators cannot  be  found  readily  by  inspection,  use  the  fol- 
lowing method : 

1.    Find  the  least  common  denominator :  ^^,  ^^,  y"^,  |J. 

Model:      2)>24     50      72     80  Find  the  least  common  multi- 

pie  of  24,  50,  72,  80.  Cancel  24 
as  it  is  a  factor  of  72.  Select  a 
prime  number  that  is  a  factor  of 
two  or  more  of  the  remaining 
5  9  2  numbers.  Divide  the  multiples 
of  this  number  by  the  number  used  as  a  divisor,  and  write  the 
quotients  and  the  numbers  that  are  not  exactly  divisible  as  shown 
in  the  model.  Continue  the  division  until  no  two  numbers  bro\ight 
down  have  a  common  factor.  The  product  of  the  several  divisors 
and  numbers  remaining  is  the  least  common  multiple  of  the 
denominators. 

1.  C.  m.=2x2x  2x  5x  Find  the  least  common  multi- 

R  w  Q  vx  o      oc*c\f\  pl®  of  the  same  numbers  by  the 

method  explained  m  Sec.  137. 


2) 

25 

36  40 

2) 

25 

18  20 

5) 

25 

9  10 

MULTIPLICATION  AND  DIVISION  113 

MULTIPLICATION  AND  DIVISION  OF  FRACTIONS 
151.  Multiplying  a  Fraction  by  an  Integer. 

1.  In  the  fraction  |,  which  term  tells  the  number  of 
equal  parts  into  which  the  unit  has  been  divided  ?  How 
many  of  the  equal  parts  are  expressed  in  the  fraction  ? 
Write  a  fraction  expressing  twice  as  many  equal  parts. 

2.  Draw  a  diagram  to  show  what  part  of  a  mile  is  ex- 
pressed in  I  mi.  Show  the  part  that  represents  f  mi. 
Compare  the  part  |  mi.  with  the  part  |  mi. 

3.  What  is  the  sum  of  |,  |,  and  |  ?     3  times  |  =  f  • 

4.  Write  I  four  times  as  an  addend  and  find  the  sum. 
State  how  the  sum  was  found. 

5.  If  I  is  written  five  times  as  an  addend,  what  is  the 
sum  ?     If  f  is  multiplied  by  5,  what  is  the  product  ? 

6.  State  how  a  fraction  may  be  multiplied  by  a  whole 
number.  Compare  the  results  thus  obtained  with  the  re- 
sults obtained  by  addition. 

7.  Multiply.  Reduce  all  products  to  their  simplest 
forms:  |  by  5;  |  by  3  ;  f  by  6 ;  f  by  7 ;  |  yd.  by  4. 

8.  If  I  is  multiplied  by  3  by  multiplying  the  numera- 
tor by  3,  the  result  will  not  be  in  its  lowest  terms.  Why? 
I  may  be  multiplied  by  3  by  dividing  the  denominator  9 
by  3.    |x3  =  f 

9.  Dividing  the  denominator  of  a  fraction  by  a  whole 
number  has  what  effect  upon  the  value  of  the  fraction  ? 

10.    Multiply  I  by  12. 

As  the  factor  4  is  common  to  both  12  and  8,  it  is  canceled  before  mul- 
tiplying. Canceling  4  in  12  leaves  3 ;  canceling  4  in  8  leaves  2.    3  times  | 

AR.  —  8 


114  FRACTIONS 

152.   Oral  Exercises. 

Solve  each  in  the  shortest  way  : 


a 

h 

c                  d 

e 

1.   1    x5 

Hx5 

f  i  X  10        i|  X  9 

Hx6 

2.   1    x4 

i%x3 

I|x7          -Hx8 

i|xl2 

3.    f    x3 

Ax7 

11  X  8          If  X  3 

Bx6 

4.    1    x5 

i|x4 

«x6          11x7 

Mx4 

5.    1    x20 

1    xl2 

41x24        1    xl6 

If  x48 

6.   1    x24 

i^.x7 

^^  X  30        f    X  42 

Iix3 

7.   11x18 

f    x5 

t\x8          ||x16 

}fx28 

8.    If  XIO 

i\x5 

ft  X  75        If  x  24 

Hx60 

9.   ^x36 

H  X  28 

Mx7          /,x8 

fix  17 

153.   Written  Exercises. 

1.    Multiply 

■43|by  8. 

Model:      43| 

o 

^            First,  multiply  |  by  8.    Next,  multiply  43  by  8. 

Add  the  products. 
344 

350 

Solve.     Perform  the  cancellation  and  reductions  with- 

out the  use  of  a  pencil : 

a 

h 

c                  d 

e 

2.   47f  x5 

64f  x7 

82fxl4    749i-Jx5 

708f  x  30 

3.    68fxT 

74f  x9 

65-1  x  15    896i|  x  7 

580f  J  X  15 

4.    96fx3 

59f  x6 

94f  X  12    780i|  X  3 

496Jix48 

5.    78fx9 

76^x9 

70f  X  18      973f  X  9 

573|  X  25 

6.   56fx9 

38|-x8 

27f  X  24      587|  x  2 

609f  X  42 

7.    Write  ten  mixed  numbers  and  multiply 

them  by  in- 

tegers. 

MULTIPLICATION  AND  DIVISION  115 

154:,  Multiplying  an  Integer  by  a  Fraction. 

1.  What  is  the  meaning  of  4  ft.  x  2?  of  4  ft.  x  1?  of 
4  ft.  X  I?  Name  the  multiplicand  and  the  multiplier  in 
each,  and  tell  what  each  shows. 

2.  4  ft.  X  I  is  the  same  as  J  of  4  ft.  How  may  |^  of  a 
number  be  found?  How  may  ^  oi  sl  number  be  found? 
When  you  know  what  J  of  a  number  is,  how  can  you  find 
J  of  the  number? 

3.  Show  with  objects  what  is  meant  by  |  of  9  things ; 
of  12  things  ;  of  6  things. 

4.  I  of  24  yd.  means  5  of  the  6  equal  parts  of  24  yd. 
Draw  a  line  to  represent  24  yd.  Divide  it  into  6  equal 
parts.     Show  the  part  that  represents  f  of  24  yd. 

5.  Show  by  a  diagram  what  is  meant  by  |  of  12  in. ; 
by  I  of  1  mi. ;  by  |  of  6  mi. ;  by  |  of  10  mi. 

6.  How  many  thirds  of  18  are  equivalent  to  18?  Are 
I  of  18  more  or  less  than  18?  If  18  is  multiplied  by  |, 
will  the  answer  be  more  or  less  than  18?     Why? 

7.  Read  each  of  the  following,  name  the  multiplicand 
and  the  multiplier  in  each,  and  tell  what  each  shows : 
120  X  f  ;  16  yr.  X  f  ;  25  mi.  x  f  ;  18  mo.  x  | ;  24  lb.  x  f . 

8.  Compare  |  of  120  with  ^  of  3  times  1 20.  Compare 
■|  of  25  mi.  with  ^  of  4  times  25  mi. 

9.  f  of  8  ft.  is  the  same  as  J  of  ft.     |  of  $5  is 

the  same  as  ^  of  $ . 

10.  Show  by  a  diagram  that  |  of  1  yd.  is  the  same  as 
i  of  3  yd. 

11.  5  divided  by  7  may  be  indicated  IJS,  or  ^.     Indi- 
cate ^  of  3 ;  J  of  2  ft.  ;  J  of  5  mi. ;  |  of  5  mi. 

12.  The  products  of  -^  x  18  and  of  18  x  |  are  the  same. 


116 


FEACTIONS 


155.    Written  Exercises. 
1.    Multiply  36  by  J|. 


3 


11 


Model  :     ^^  x  ^  =  .^  =  16  J. 


12  is  a  factor  common  to 
36  and  24.  Cancel  the  con*- 
mon  factor.  3  times  11  is 
33;  -8^=16i. 


Solve  : 

a 

h 

c 

rf 

e 

2.    30  x| 

Sxif 

25  xf 

144  xf 

100  xf 

3.    48xiJ 

Txf 

30x^5^ 

54  X  J 

100  xf 

4.    36  X  ^ 

8xi\ 

27  X  il 

60  Xt^ 

100  x| 

5.    21  xf 

9xi| 

45  xf 

16xJ| 

100  xf 

6.     4xf 

6xf 

72x|J 

86  x| 

100  xf 

156.  Written  Exercises. 

1.    Multiply  845  by  4f . 

Model  :     845 


f  of  845  is  120f ;    ^  of  845  are  3  times  120f,oP 
^T  362f    4  times  845  are  3380.     Add  the  products. 
120f   (I  of  845) 


3621 

(f  of  845) 

3380 

(4  times  845) 

3742| 

Multiply  : 

a 

h 

c 

d 

2. 

60x8f 

64  X  45f 

827  X  47f 

801  X  84f 

3. 

36x9f 

81x47^2^ 

459  X  75f 

153  X  46f 

4. 

55  X  8^^ 

72  X  67f 

693  X  68| 

360x48| 

5. 

27x6| 

96  X  87-1 

745  X  47| 

578x96f 

6. 

33x4| 

48  X  541 J 

584  X  37| 

609x24f 

7. 

45x8f 

75  X  49^ 

144  X  35{j. 

586  x  27| 

MULTIPLICATION  AND  DIVISION  117 

157.  Dividing  a  Fraction  by  an  Integer. 

1.  Divide  12  by  3.  Find  ^  of  12.  -l  of  a  number  may 
be  found  by  dividing  the  number  by  3.  State  how  |^  of  a 
number  may  be  found. 

2.  Draw  a  line  12  inches  long.  Show  -^^  of  the  line. 
Show  \  of  -j^  of  the  line.  Divide  -f^  of  the  line  into  3 
equal  parts.  How  does  each  of  these  parts  compare 
with  I  of  -^^  of  the  line?     J  of  -^^  is  the  same  as  -^^  -f-  3. 

3.  What  is  \  of  I?  of  I?  of  if?  of  1^? 

4.  State  how  a  fraction  may  be  divided  by  an  integer, 
when  the  numerator  is  exactly  divisible  by  the  integer. 

5.  Solve:  f^-^3;  l|-^7;  lf-^6;  ||-^8;  ^f-^5. 

6.  Draw  a  line  12  in.  long.  Divide  it  into  12  equal 
parts.  Each  part  is  -^^  of  the  whole  line.  Divide  each 
part  into  3  equal  parts.  Each  of  these  smaller  parts  is  what 
part  of  the  entire  line  ?  To  divide  -^  of  the  line  into  3 
equal  parts,  each  of  the  8  parts  must  be  divided  into  3 
equal  parts  and  ^  of  these  taken.  Show  that  J  of  -^  of 
the  line  is  ^  of  the  line.  This  result  may  be  found  by 
multiplying  the  denominator  of  ^^2  t)y  3. 

7.  Divide  \^  by  4  by  dividing  the  numerator  by  4  ;  by 
multiplying  the  denominator  by  4,     Compare  the  results. 

158.  Oral  Exercises. 

Divide,  using  the  shortest  method  for  each : 

1.  -rV^yS  5.  if  by  2  9.  |T.  by2 

2.  f^bylO  6.  If  by  6  10.  $tby7 

3.  I  by  6  7.  if  by  7  11.  1|  mi.  by  5 
4-  M  by  8  8.  2§  ft.  by  3  12.  f  da.  by  3 


118  FRACTIONS 

159.  Written  Exercises. 

1.    Divide  65|  bj  8. 

TVT  .  ^S^         ^  ^^  contained  in  65  eight  times,  with  1 


J.M.I 

^^^^-      8)65| 

over ;  ^  of  If  is  ^  of 

h  or  ij. 

2. 

37^^  7 

5 
325|^4 

c 
1321  H- 5 

3. 

62|-j-9 

423f^6 

836f^4 

4. 

87f-f-8 

7561^8 

456^  ^  2 

5. 

46|-^9 

637|-f-7 

3871-*- 3 

6. 

90-1-  ^  8 

436^^9 

7261^14 

7. 

Divide  645f  by 

24  ;  1645|  by  3^ 

; ;  195|  by  27. 

8. 

Divide  347f  by 

46;    73|byl4; 

4721  by  ^c>. 

160,  Written  Exercises. 

1.  A  carpenter  sawed  a  board  9  ft.  4  in.  (9 J  ft.)  long 
into  4  equal  parts  to  make  shelves  for  a  bookcase.     How 

long  was  each  shelf? 

2  ft.  4  in. 
Or,  divide  9  ft.  4  in.  by  4  thus:  4)9  ft.  4  in.     ^  of  8  ft.  is  2  ft.; 
I  of  16  in.  (1  ft.  4  in.)  is  4  in.    Ans. :  2  ft.  4  in. 

2.  The  perimeter  of  a  square  flower  bed  is  14  ft.  8  in. 
(14|  ft.).     How  long  is  each  side? 

3.  If  a  train  travels  at  an  average  rate  of  45  mi.  per 
hour,  how  far  will  it  travel  in  4  hr.  45  min.  (4|  hr.)  ? 

4.  At  5  ^  per  pound,  how  much  will  11|  lb.  of  sugar  cost  ? 

5.  In  a  magazine  of  160  pages,  45  pages  were  devoted 
to  advertisements.  What  part  of  the  magazine  was  de- 
voted to  advertisements? 

6.  Two  boys  caught  8  fish.  The  combined  weight  of 
the  fish  was  lOJ  lb.     What  was  their  average  weight? 


MULTIPLICATION  AND  DIVISION  119 

161.   Multiplying  a  Fraction  by  a  Fraction. 

1.  Show  by  a  diagram  or  with  objects  what  is  meant 
by  f  of  6  ft. ;  by  f  of  1  ft. ;  by  f  of  i  ft.  |  of  ^  ft.  is 
what  part  of  1  ft.  ? 

2.  Draw  a  line  and  divide  it  into  5  equal  parts.  What 
is  each  part  called  ?  Show  ^  of  one  of  these  parts.  |  of 
^  of  the  line  is  what  part  of  the  line  ? 

3.  Draw  a  line  and  divide  it  into  4  equal  parts.  Show 
■J  of  one  of  these  parts.  |^  of  J  of  a  line  is  what  part  of 
the  line  ? 

4.  Draw  a  line  and  divide  it  into  3  equal  parts. 
Show  ^  of  one  of  these  parts.  Show  f  of  one  of  these 
parts.     -J^  of  |-  of  the  line  is  what  part  of  the  entire  line  ? 

5.  How  much  is  I  of  6  da.  ?  J  of  6  sevenths  ? 

6.  How  much  is  -J  of  ^g-  ?  of  -^j  ?  of  1|  ? 

7.  When  you  know  what  |^  of  a  number  is,  how  can 
you  find  |  of  the  number  ?  When  you  know  what  ^ 
of  a  fraction  is,  how  can  you  find  |  of  the  fraction  ? 

8.  State  how  you  would  find  :|  of  i| ;  J  of  1| ;  J  of  if. 

9.  State  how  you  would  find  J  of  f  ;  |  of  | ;  ^  of  | ; 
I  off. 

10.  Divide  rectangles  to  show  |  of  | ;  |  of  | ;  |  of  |^ ; 
f  of  ih 

11.  Show  by  a  diagram  that  J  of  |-  is  ■^.     Since  |  of  J 
is  ^2'  f  o^  i  ^s  ^^w  many  times  -^^  ^ 

12.  Divide  rectangles  to  show  that  ^  of  J  is  equivalent 
to  J  off 

13.  Divide  rectangles  to  show  |  of  |^  and  |^  of  | ;  f  of  f 
and  f  off;  I  of  1  audi  off. 


120  FRACTIONS 

14.  Draw  a  line.  Show  ^  of  the  line.  Show  |  of  the 
line.  Show  J  of  ^  of  the  line.  What  part  of  the  line  is 
J  of  I  of  the  line  ?  Show  J  of  |  of  the  line.  J  of  f  of 
the  line  is  what  part  of  the  line  ?  Show  |  of  |  of  the 
line,     f  of  |-  of  the  line  is  what  part  of  the  line? 

l—J . 1 . . 1 U—i \ . . I         .         ■         I 

J  of  ^  of  the  line  =  ^  of  the  line. 
L-j—j I I I I I I I I I \ ■      ■      1 

I  of  the  line  =  -^  of  the  line. 
I  of  f  of  the  line  =  ^^  of  the  line.     ^  of  f  =  ^^. 
1 1 I ^       '      ■ \      »      *      \      t I I '       ^ 

V > 

I  of  I  of  the  line  =  -^  of  the  line,     f  of  |  =  i^* 

15.  Draw  a  line.  Show  |  of  the  line.  Show  |  of  the 
line.  Show  J  of  ^  of  the  line.  J  of  J  of  the  line  is  what 
part  of  the  line  ?  Show  J  of  f  of  the  line.  |  of  f  of  the 
line  is  what  part  of  the  line  ?  Show  |  of  |  of  the  line. 
I  of  I  of  the  line  is  what  part  of  the  line  ? 

16.  Finding  |-  of  a  number  is  the  same  as  multiplying 
the  number  by  J.  Finding  f  of  a  number  is  the  same  as 
multiplying  the  number  by  |.  Examine  the  illustrations 
under  Ex.  14,  and  tell  how  J  of  J  of  a  number  is  found; 
J  of  I  of  a  number  ;  |-  of  |  of  a  number. 

17.  Since  J  of  ^  is  ^,  J  of  f  is  ^^  and  |  of  f  is  ^. 
In  each  case  the  product  of  the  numerators  is  the  numer- 
ator of  the  answer,  and  the  product  of  the  denominators  is 
the  denominator  of  the  answer. 

To  multiply  a  fraction  hy  a  fraction,  multiply  the  numera- 
tors for  the  numerator  of  the  product  and  the  denominators 
for  the  denominator  of  the  product.  Before  midtiplyitig^ 
cancel  factors  common  to  both  terms. 


MULTIPLICATION  AND  DIVISION  121 

162.   Written  Exercises. 

1.    Multiply  fl  by  f . 

Model  : 
3        1  8  is  a  common  factor  of  24  and  8.     5  is  a  com- 


?^,^_3 

nion  factor  of  5  and  25. 

Cancel  these  common 

n^"^ 

factors  and 

multiply. 

5      1 

Solve.     Before  multiplying,  cancel 

common 

factors : 

a 

b 

c 

d 

e 

2-       fxf 

f|xf 

II  X  11 

h^xi 

Mxl! 

3-       ixf 

II  X  il 

f>'l 

|x| 

Ifxi| 

4.       |X| 

\\  X  \ 

l>-f 

Jxii 

^Xt^j 

5.       fxf 

if  X  1 

¥xf 

^x^ 

Wx| 

163.  Written  Exercises. 

Change  the  mixed  numbers  to  improper  fractions  and 
solve  : 


a 

6 

c 

d 

1. 

Hx^ 

fx2i 

2|x4f 

l|x3i 

2. 

4ix| 

|x6|. 

6|x4| 

4Jx2J 

3. 

Sfxf 

|x4| 

7|x4| 

6|x4| 

4. 

5^x1 

T%x5i 

9fx6| 

8|x2J 

164.  Written  Exercises. 

Review  Sees.  152,  153,  and  156. 

1.    Multiply  45f  by  |. 

Model  :         45|  ^  of  45§  is  7ii;  f  of  45f  =  5  times  7H..  or 

5       38^1,. 

6 

1{1    (^of45|) 
5 
38 J^    (iof45|) 


122  FRACTIONS 

Solve  without  reducing  the  mixed  numbers  to  improper 
fractions  : 

a  b  c  d  e 

2.  34|x|     84fxf    546|    x|     654f    xf      840f  x  f 

3.  ISfxf  55|xf  385|    xf  235|    xf  468f  x  | 

4.  72fx|  40|x|  463J    xf  900,9^  x  |  479|  x  | 

5.  48fx|  38Jx|  847A    x|  783f    x  J  673f  x  f 

6.  96fxf  94^  X  I  170^\x|  680^  x  ^^  574|  x  | 

7.  481  x^  63Jx|  4311  x|  598f  x^  650J  x  | 

165.  Written  Exercises. 

1.    Multiply  349f  by  3f , 

Model  :      349|  First  multiply  349f  by  f .    Next  multiply  349| 

ga       by  3.     Add  the  products. 


mi 

3 

(I  times  349f) 

209t 
1049 

12581 

(1  times  349f) 
(3  times  349f) 

Solve  without  reducing  the  mi^ed  numbers  to  improper 
fractions  : 

abed 
2.    645fx6f      584^^  x4f      963J  x  7J      642f    x  7^ 

3.    867fx5f 

982f    x8f 

333ix6§ 

789^  X  7f 

4.    694fx7i 

648|    x5J 

78Hx9f 

537^  X  6  J 

5.    748^  x5f 

457f    x7| 

450f  x  7f 

521^    x2| 

6.    384fx4f 

926|   x2f 

467J  X  9J 

830|    x4f 

7.    412|x6| 

726|    x8J 

940^  X  3f 

590^  X  7J 

8.    240|x4f 

948|    xl^ 

640^  X  7f 

810f    x3f 

MULTIPLICATION   AND  DIVISION  123 

166.  Review  Questions. 

1.  What  is  a  proper  fraction?  an  improper  fraction? 
a  mixed  number  ? 

2.  Write  5  proper  fractions;  5  improper  fractions; 
5  mixed  numbers. 

3.  What  is  a  fractional  unit?  How  many  fractional 
units  are  expressed  in  |? 

4.  What  is  meant  by  a  factor  of  a  number?    Illustrate. 

5.  What  is  meant  by  a  multiple  of  a  number  ?  Illustrate. 

6.  When  is  a  fraction  said  to  be  in  its  lowest  terms? 
Write  five  fractions  that  are  in  their  lowest  terms. 

7.  How  may  a  fraction  be  reduced  to  its  lowest  terms  ? 
Write  five  fractions  and  reduce  them  to  their  lowest  terms. 

8.  How  may  an  improper  fraction  be  reduced  to  a 
whole  or  a  mixed  number?  Write  five  improper  fractions 
and  reduce  them  to  whole  or  mixed  numbers. 

9.  How  may  a  mixed  number  be  changed  to  an  im- 
proper fraction?  Write  five  mixed  numbers  and  change 
them  to  improper  fractions. 

10.  What  effect  upon  the  value  of  a  fraction  has  multi- 
plying or  dividing  both  terms  of  the  fraction  by  the  same 
number?     Illustrate. 

11.  What  is  cancellation?     Illustrate. 

12.  State  two  ways  in  which  a  fraction  may  be  multi- 
plied by  an  integer.  Illustrate.  When  a  proper  fraction 
is  multiplied  by  an  integer,  is  the  product  greater  or  less 
than  the  multiplicand?  Why?  Is  the  product  greater  or 
less  than  the  multiplier?     Why?     Illustrate. 

13.  When  an  integer  is  multiplied  by  a  proper  fraction, 
is  the  product  greater  or  less  than  tlie  multiplicand? 
than  the  multiplier?     Why?     Illustrate. 


124  FRACTIONS 

167.  Dividing  by  a  Fraction, 

1.  Draw  a  line  4  ft.  long.  Make  a  measure  J  ft. 
long.  Apply  this  measure  to  the  line.  How  many  times 
must  the  measure  J  ft.  be  applied  to  measure  a  4-ft.  line? 

2.  After  finding  how  many  times  the  measure  |  ft. 
must  be  applied  to  measure  1  ft.,  how  may  you  find,  with- 
out performing  the  actual  measurement,  how  many  times 
the  measure  must  be  applied  to  measure  a  4-ft.  line  ? 

3.  Repeat  Exs.  1  and  2  above,  using  a  J-ft.  measure  to 
measure  a  6-ft.  line. 

4.  As  the  measure  ^  ft.  must  be  applied  2  times  to 
measure  1  ft.,  to  measure  any  given  number  of  feet  it 
must  be  applied  as  many  times  2  as  the  number  of  feet  to 
be  measured.  To  measure  24  ft.,  it  must  be  applied  24 
times  2,  or  48  times.  How  many  times  must  the  measure 
J  ft.  be  applied  to  measure  6  ft.  ?   10  ft.  ?   12  ft.  ? 

5.  To  measure  a  line  J  ft.  long,  the  measure  J  ft. 
must  be  applied  J  times  2,  or  J  times.  That  is,  one  half 
of  the  measure  must  be  applied. 

6.  The  expression  J  ft.  -4-  J  ft.  indicates  that  a  line  J 
ft.  in  length  is  to  be  measured  by  a  measure  J  ft.  in 
length.  What  is  meant  by  each  of  the  expressions: 
6ft.^Jft.?   |ft.+Jft.?  Jjft.-i-Jft.? 

7.  Draw  a  line  J  ft.  long.  Determine  how  many 
times  each  of  the  following  measures  must  be  applied  to 
measure  it:  J  ft.,  J  ft.,  ^  ft. 

8.  The  measure  |  ft.  must  be  applied  how  many  times 
to  measure  a  1-ft.  line?  1J=  J.  It  must  be  applied  | 
times.  How  many  times  must  it  be  applied  to  measure  a 
4-ft.  line?  a  6-ft.  line?  a  15-ft.  line?  What  is  meant  by 
the  expression  18  ft.  -*-  f  ft.  ? 


MULTIPLICATION   AND  DIVISION  125 

9.  If  a  |-ft.  measure  must  be  applied  |  times  to 
measure  a  1-ft.  line,  to  measure  a  |-ft.  line  it  must  be  ap- 
plied I  times  |,  or  |  times. 

10.  To  measure  a  1-ft.  line,  a  2-ft.  measure  must  be  ap- 
plied I  time.  That  is,  one  half  of  the  measure  must  be 
applied.  How  many  times  must  the  measure  3  ft.  be  ap- 
plied to  measure  a  1-ft.  line? 

11.  Determine  how  many  times  the  measure  in  each 
must  be  applied  to  measure  1  ft.,  and  solve  each: 
5  ft.-^^  ft.;  3  ft.^J  ft.;  |  ft. -^  2  ft.;  J  ft. -i- 3  ft.; 
4  ft.  -^  1  ft. 

12.  To  measure  a  1-ft.  line,  the  measure  ^  ft.  must  be 
used  2  times.  2  is  the  reciprocal  of  J.  To  measure  a 
1-ft.  line,  the  measure  2  ft.  must  applied  J  times.  \  is  the 
reciprocal  of  2. 

13.  The  reciprocal  of  4  is  | ;  of  3  is  J ;  of  ^  is  3 ;  of  f 
is  |;  of  1^  is  |.  If  I  is  used  as  a  measure  to  measure  1, 
the  quotient  is  |.     Multiply  |  by  |.     The  product  is  1. 

14.  When  the  product  of  two  numbers  is  1,  the  num- 
bers are  said  to  be  reciprocals  of  each  other. 

15.  The  reciprocal  of  the  number  used  as  divisor  shows 
the  number  of  times  the  divisor  is  contained  in  a  unit, 
thus :  The  divisor  ^  is  contained  in  1  three  times.  The 
divisor  J  is  contained  in  1  f  times.  Hence  the  following 
rule : 

To  divide  hy  a  fraction^  miultiply  the  reciprocal  of  the  di- 
visor hy  the  dividend. 

16.  What  is  the  reciprocal  of  each:  |?  ^?  |^?  y^^?  ^? 

17.  Compare  the  terms  of  a  fraction  with  the  terms  of 
the  reciprocal  of  the  fraction.  When  the  terms  of  a  frac- 
tion are  interchanged,  the  fraction  is  said  to  be  inverted. 


56                                   Fl 
168.  Written  Exercises 

MCI 

IONS 

1.    Divide  6|  by  f . 

3 

Model:      6| -f- ^  =  ?I  > 

6      i 

^5 

Solve :                       2 

2.    3|  yd. -1- f  yd. 

10. 

6f^6f 

18. 

H^f 

3.    5iyd.  ^|yd. 

11. 

4|  ^  6| 

19. 

3J^2J 

4.    6f  wk.  -i-  f  wk. 

12. 

5|-2,V 

20. 

81J^9^ 

5.    8|  yd.  H-  1  yd. 

13. 

7f^8J 

21. 

m^^^ 

6.    |7f-i-|| 

14. 

f^l 

22. 

16|^14f 

7.    3y\  in.  ^  1  in. 

15. 

1%-^T% 

23. 

6|^12| 

a  8|-f-4| 

16. 

A^l 

24. 

7J-^6|i 

9.   7^^5f 

17. 

A^? 

25. 

24|^8 

26.  Divide  100  by  331 ;  by  66f ;  by  371;  by  87 J. 

27.  Divide  if  by  4 ;  if  by  5 ;  316  J  by  8  ;  435|  by  27. 

28.  Multiply  635f  by  8;  315f  by  8;  80f  by  9. 

29.  Add  8f,  4|,  31  6^,  8^. 

30.  Take  32f  from  96|.     From  80|  take  19f . 

31.  Divide  8.125  by  .04;  180.40  by  .05;  725  by  1.25. 

32.  Multiply  3.1416  by  4|;  .7854  by6J;  2150.42  by 
60|. 

169.  Oral  Exercises. 

1.  Divide  each  by  100 :  |43,  3.14,  60.75,  .9,  2000. 

2.  Add  J  and  \ ;  J  and  \  ;  \  and  \.  State  a  short 
method  of  getting  the  sum  of  two  fractions  whose  nu- 
merators are  1  and  whose  denominators  are  prime  to  each 
other. 


MULTIPLICATION  AND   DIVISION  127 

170.  1.    How  many  strips  of  carpet  1  yd.  wide  will  it 
take  to  cover  a  room  7  yd.  wide  ? 

2.  How  many  strips  of  carpet  |  yd.  wide  will  it  take 
to  cover  a  room  9  yd.  wide  ?  6  yd.  wide  ?  3  yd.  wide  ? 

3.  Draw  a  diagram  of  a  room  24  ft.  long  and  18  ft. 
wide.  Show  on  the  diagram  the  number  of  strips  of 
carpet  |  yd.  wide  that  are  needed  to  cover  the  floor,  the 
strips  running  lengthwise  of  the  room, 

4.  What  is  the  length  in  yards  of  each  strip  (Prob. 
3)?  How  many  yards  of  carpet  are  needed  to  cover 
the  room,  making  no  allowance  for  matching  the  strips  ? 

5.  At  75^  per  yard,  how  much  will  it  cost  for  carpet 
for  a  room  28  ft.  long  and  18  ft.  wide,  the  carpet  being 
27  in.  wide,  the  strips  running  lengthwise  of  the  room  ? 

6.  How  many  ribbons  each  -|  yd.  long  can  be  made 
of  8  yd.  of  ribbon?  of  12  yd.?  of  18  yd.?  of  6  yd.? 

7.  At  5/  a  pound,  how  many  pounds  of  sugar  can 
be  bought  for  40^? 

8.  How  many  pounds  of  sugar  can  be  bought  for  1 5, 
at  4^^  a  pound  ?  at  4|^  a  pound  ?  at  5|  ^  a  pound  ? 

9.  How  many  strips  of  matting  42  in.  (IJ  yd.)  wide 
will  it  take  to  cover  a  room  21  ft.  (7  yd.)  wide? 

10.  If  a  certain  lamp  consumes  |  pt.  of  oil  each  even- 
ing, how  long  will  a  gallon  of  oil  last  ?   5  gal.  ? 

11.  What  part  of  1  yd.  is  1  ft.?  30  in.  ?  32  in.  ?  27  in.  ? 

171,  1.    How  many  times  must  2|  T.  be  written  as  an 
addend  so  the  sum  of  the  column  will  be  24  T.  ? 

2.  If  a  boy  earns  <l  |  a  day,  in  how  many  days  will  he 
earn  |15  ? 


128  FRACTIONS 

3.  1  yd.  of  cloth  will  cost  how  many  times  the  cost  of 
I  yd.? 

4.  If  a  dealer  charges  f  6  for  |  T.  of  coal,  what  is  the 
price  of  the  coal  per  ton  ? 

5.  Find  the  cost  of  1  yd.  of  lace  if  2J  yd.  cost  75  ^. 

6.  At  $  1 J  per  yard,  what  will  be  the  cost  of  8 J  yd.  of 
silk? 

7.  Find  the  area  of  a  rectangle  H^  in.  by  6|  in.;  of  a 
square  whose  side  is  2|  ft. 

8.  Hqw  many  pounds  of  meat  at  12 J^  a  pound  can  be 
bought  for  75^? 

9.  If  3|  lb.  of  coffee  are  sold  for  $  1,  how  many  pounds 
can  be  bought  for  $  6  ?  for  1 3  ?  for  $  9  ?  for  $  12  ? 

10.  If  1  lb.  of  tea  costs  $  |,  how  many  pounds  can  be 
bought  for  f  2f  ?  f or  $  6  ?  for  $  12  ? 

11.  A  tailor  used  2|  yd.  of  cloth  for  each  pair  of  trou- 
sers.    How  many  pairs  can  be  made  from  22  yd.  ? 

12.  Change  to  feet :  6  in.  ;  9  in. ;  10  in.  ;  3  in. ;  4  in. 

13.  Change  to  inches  :  |  ft.  ;  |  ft.  ;   |  ft.  ;  J  f t. ;  ^  ft. 

14.  Change  to  months  :  J  yr.  ;  J  yr. ;  §  yr. ;  |  yr.  ;  f  yr. 

15.  The  atmosphere  presses  equally  in  all  directions 
with  a  pressure  of  about  15  lb.  to  the  square  inch.  Find 
the  pressure  on  the  top  of  your  desk. 

16.  If  the  circumference  of  the  wheel  of  a  bicycle  is  6^ 
ft.,  how  many  times  will  the  wheel  turn  in  going  1  mi.? 

17.  The  cost  of  laying  a  concrete  sidewalk  at  11 J  ^  per 
square  foot  was  $34.50.  Find  the  area  of  the  sidewalk. 
If  the  walk  was  6  ft.  wide,  how  long  was  it  ? 


MULTIPLICATION  AND  DIVISION  129 

172.    Review  Exercises. 

1.  Find  the  perimeter  of  a  rectangle  8  ft.  6  in.  wide 
and  12  ft.  9  in.  long.     Find  its  area. 

2.  Find  the  number  of  square  feet  of  blackboard  sur- 
face in  the  schoolroom. 

3.  The  diameter  of  a  cylindrical  tank  is  6.5  ft.  Find 
its  circumference,     (circum.  =  diam.  x  3|.) 

4.  A  farmer  asked  his  two  boys,  George  and  Frank,  to 
figure  out  the  number  of  posts  necessary  to  build  a  fence 
28  rd.  long,  the  posts  to  be  placed  J  rd.  apart.  George 
said  it  would  take  56  posts,  and  Frank  said  it  would  take 
67.     Was  either  boy's  answer  correct  ? 

5.  The  farmer  (Prob.  4)  asked  the  boys  to  find  the 
number  of  posts  necessary  to  build  a  fence  around  a  gar- 
den 6  rd.  by  8  rd.,  the  posts  to  be  placed  J  rd.  apart. 
Both  boys  said  it  would  take  57  posts.  Was  the  answer 
correct  ? 

6.  If  a  man  had  60  sheep  and  sold  |  of  them,  how 
many  did  he  sell  ?     How  many  did  he  have  left  ? 

7.  There  are  640  acres  in  1  square  mile.  How  many 
are  there  in  |  of  a  square  mile  ? 

8.  If  f  of  the  distance  between  two  cities  is  15  mi., 
how  far  apart  are  the  cities  ? 

9.  Mary's  age  is  12  years.  She  is  |  as  old  as  Ethel. 
How  old  is  Ethel  ? 

10.  The  cost  of  15  T.  of  hay  was  $112.50.     What 
was  the  cost  per  ton? 

11.  At  75^  a  yard,  how  many  yards  of  cloth  can  be 
bought  for  $15? 


130  FRACTIONS 

12.  Find  the  value  of  the  potato  crop  on  50  acres,  if 
the  average  yield  is  125  sacks  to  the  acre  and  the  potatoes 
are  worth  il.20  per  sack. 

13.  Draw  three  dials,  as  in  Sec.  97,  and  fix  the  hands 
to  read  84,500  cu.  ft.  ;  32,800  cu.  ft.  ;  47,000  cu.  ft. 

14.  Is  the  height  of  any  mountain  given  in  your  geog- 
raphy text  ?  If  so,  express  the  height  of  some  mountains 
in  miles. 

15.  Draw  a  line  20  in.  long.     Test  it  with  a  ruler. 

16.  If  the  height  of  your  schoolroom  is  12  ft.  6  in.,  find 
the  distance  from  the  highest  point  on  your  desk  to  the 
ceiling. 

17.  Find  the  area  of  your  desk  top. 

18.  Find  the  area  of  a  blackboard  in  your  schoolroom. 

19.  Draw  a  diagram  to  show  the  ratio  of  2  in.  to  4  in. ; 
of  4  in.  to  6  in. ;  of  8  in.  to  12  in. 

20.  What  number  expresses  the  ratio  of  3  in.  to  6  in.? 
of  6  in.  to  3  in.?  of  8  in.  to  4  in.?  of  4  in.  to  12  in.? 

21.  Express  the  value  of  5|  in  the  fractional  unit  J  ;  of 
4f  in  the  fractional  unit  ^  ;  of  8|  in  the  fractional  unit  J. 

22.  From  a  piece  of  cloth  containing  22J  yd.  a  tailor 
used  17|  yd.     How  many  yards  remained  in  the  piece? 

23.  .  How  much  heavier  is  Mary  than  Ethel,  if  Mary 
weighs  101 J  lb.  and  Ethel  weighs  92|  lb.?  Find  the  com- 
bined weight  of  the  two  girls. 

24.  What  effect  upon  the  value  of  a  fraction  has  multi- 
plying or  dividing  both  terms  by  the  same  number? 
Illustrate,  using  ^. 

25.  Show  by  a  diagram  that  f  of  3  ft.  is  the  same  as  J 
of  2  times  3  ft. 


MULTIPLICATION   AND   DIVISION  131 

173.  Finding  what  Fraction  One  Number  is  of  Another. 

1.    What  fraction  of  a  dollar  is  33 J^? 

Model:     33^^  is —r^ of  a  dollar.     Performing  the  indicated  divi- 
sion :  33^  H-  100,  or  ife  x  ik  =  i. 

To  find  what  fraction  one  number  is  of  another^  take  the 
number  denoting  the  part  for  the  numerator  and  the  number 
denoting  the  whole  for  the  denominator.  Express  the  result 
in  its  simplest  form. 

What  fraction  of : 


2. 

8  is  5? 

9. 

100  is  75? 

16. 

100  is  121? 

3. 

15  is  10? 

10. 

100  is  60? 

17. 

100  is  37|  ? 

4. 

20  is  24? 

11. 

100  is  125? 

18. 

100  is  1371? 

5. 

16  is  10? 

12. 

100  is  175? 

19. 

100  is  1331? 

6. 

36  is  30? 

13. 

100  is  120? 

20. 

100  is  66f  ? 

7. 

100  is  25? 

14. 

100  is  40? 

21. 

100  is  166f  ? 

8. 

100  is  150  ? 

15. 

100  is  87J? 

22. 

100  is  1121? 

23.  What  fraction  of  a  dollar  is  75^?  50^?  25 i^?  121^? 
20^?  60^?  371^?  331/?  66|/?  87|/?  62|/?  80^?  70^? 
40/?  150/?  125/?  175/?  14f/  16|/?  112|/?  140/? 
6J/? 

24.  If  a  train  travels  at  an  average  rate  of  40  mi.  per 
hour,  in  what  part  of  an  hour  will  the  train  travel  15  mi.  ? 
25  mi.  ?  60  mi.  ?  75  mi.  ?  100  mi.  ? 

25.  If  a  20-acre  field  produced  $600  worth  of  wheat  in 
a  certain  year,  at  the  same  rate  what  part  of  this  amount 
would  a  10-acre  field  have  produced  ?  a  30-acre  field  ?  a  15- 
acre  field?  a  25-acre  field?  a  60-acre  field? 


132 


FRACTIONS 


DRAWING  TO  A  SCALE 

174.  1.  Lines  and  surfaces  are  frequently  represented 
by  drawings.  As  most  lines  and  surfaces  are  too  large  to 
be  drawn  full  size,  they  have  to  be  drawn  on  a  reduced  scale. 

2.  By  letting  |  in.  represent  1  ft.,  a  line  8  ft.  long 
may  be  represented  thus: 

8  ft. 


Scale  i  in.  =  1  ft. 

3.  Using  the  scale  J  in.  =  1  ft.,  represent  a  line  12  ft. 
long ;  5  ft.  long ;  20  ft.  long ;   16  ft.  long. 

4.  Using  the  scale  J  in.  =  1  ft.,  represent  a  square 
whose  side  is  4  ft.;  6  ft.;  12  ft. 

5.  Using  the  scale  J  in.  =  1  yd.,  represent  a  line  12 
yd.  in  length  and  a  line  8  yd.  in  length. 

6.  Using  a  convenient  scale,  represent  a  rectangle  12 
ft.  by  10  ft.;  a  garden  16  ft.  by  12  ft. 

7.  Using  the  scale  ^  in.  =  1  ft.,  represent  a  school 
garden  24  ft.  by  16  ft.     Find  the  area  of  this  garden. 

8.  Using  a  convenient  scale,  represent  a  sidewalk  6  ft. 
wide  and  48  ft.  long.     Find  the  area  of  this  walk. 

9.  Below  is  a  diagram  of  a  school  yard,  drawn  to 
the  scale  ^  in.  =  20  ft.  Find  the  dimensions  of  the  yard. 
Find  its  area. 


SCALE  DRAWING  133 

176.   1.    Find  the  scale  to  which  each  of  the  following 
lines  has  been  drawn : 

^      60ft 

/?     90  f I 


C 
D 

£ 


/soft 


4Sfl 
/20yd 


2.  Find  the  dimensions  of  the  floor  of  your  schoolroom. 
Using  a  convenient  scale,  draw  a  floor  plan  of  your 
schoolroom. 

3.  Find  the  dimensions  of  the  school  grounds.  Using 
a  convenient  scale,  draw  a  plan  of  the  school  grounds. 
Show  the  ground  plan  of  the  schoolhouse  properly  located 
and  in  correct  proportions. 

4.  The  relative  areas  of  the  several  oceans  are  repre- 
sented by  the  lines  below,  as  follows :  A^  Arctic ;  jB, 
Antarctic ;  (7,  Indian ;  i),  Atlantic ;  H^  Pacific.  The  num- 
bers above  the  lines  indicate  the  number  of  millions  of 
square  miles  in  each. 


23 


3± 

7/ 


5.  Using  a  convenient  scale,  represent  the  relative 
population  of  Asia,  Europe,  Africa,  North  America,  South 
America. 


184 


FRACTIONS 


d 

c  — 

e 

a 

b 


176,  1.  This  figure  represents  a  section  of  land.  Find 
the  dimensions  of  each  division ;  the  num- 
ber of  acres  in  each ;  the  cost  of  the  sec- 
tion; and  of  each  division  at  $45  per  acre. 
(Section  =  1  sq.  mi.=  640  A.) 

2.  A  field  containing  10  A.  is  40  rd. 
long.  How  wide  is  it?  Draw  it  to  a 
scale. 

3.  This  figure  represents  a  garden.  What  is  the  scale  ? 
Draw  the  plan  of  this  garden  on  a  scale  twice  as  large  as 

the  figure,  viz.  \  in.  =  l  rd. 
Divide  the  garden  into 
three  rectangles.  Find 
the  area  of  each  rec- 
tangle. Find  the  area  of 
the  garden ;  the  perime- 
ter. Find  the  cost  of 
fencing  this  garden  at 
11.75  per  rod. 

4.  Draw  to  a  scale  the  side  of  your  schoolroom  and 
locate  the  openings.  Mark  the  dimensions  on  your  drawing. 

5.  Draw  lines  to  represent  the  population  of  New  York 
City,  Chicago,  Philadelphia,  London,  and  Paris,  using  the 
same  scale  for  each. 


•g       6rd. 

4-rd. 

1 

>. 

16  rd. 

erd 

111.  City  Lots. 

1.  The  figure  on  p.  135  is  a  diagram  of  a  city  block. 
The  dimensions  are  expressed  in  feet.  Using  your  ruler, 
determine  the  scale  used  in  making  this  diagram.  Find 
the  width  of  East  Avenue. 


SCALE  DRAWING 


135 


_j 


Grove   St 


I 

0) 
1. 


too 

^ 
o 

^ 

-^ 
^ 

l\) 

Co 
Cr> 

Co 

Co 

•li^ 

Oi 

6 

7 

8 

9 

/O 

^5 

60 

40 

30 

30 

40 

East  Av. 


~^ 


186  FRACTIONS 

2.  Find  the  length  and  the  width  of  the  block. 

3.  Find  the  area  of  each  of  the  lots. 

4.  Lot  7  was  sold  at  $35  per  front  foot.     Find  the 
selling  price  of  the  lot. 

5.  At  the  same  price  per  front  foot,  what  is  the  value 
of  each  of  the  other  lots  ? 

6.  At  $45  per  front  foot,  what  is  the  value  of  Lot  6 
(60  ft.  front)  ? 

7.  Lot  1  was  sold  for  $2250.     How  much  was   this 
per  front  foot? 

8.  Lot  10  was  sold  for  $  1600.     How  much  was  this 
per  front  foot  ? 

9.  The  selling  price  of  Lot  5  was  $  750.     How  much 
was  this  a  front  foot  ? 

10.  From  its  location  in  the  block,  which  should  be 
worth  the  more  per  front  foot.  Lot  7  or  Lot  10  ?  Lot  4 
or  Lot  5  ? 

11.  Find  the  cost  of  laying  a  6-ft.  cement  sidewalk  in 
front  of  Lot  9  at  12  ^  per  square  foot. 

12.  Find  the  cost  of  laying  a  6-ft.  cement  sidewalk  in 
front  of  Lot  2  at  13  ^  per  square  foot. 

13.  Mr.  Thomas  bought  Lot  8  at  $35  per  front  foot. 
He  sold  it  for  $1200.  Did  he  gain  or  lose,  and  how 
much? 

14.  Mr.  Brown  paid  $  35  per  front  foot  for  Lots  3  and 
4.  He  sold  both  lots  for  $2500.  Did  he  gain  or  lose, 
and  how  much  ? 

15.  Mr.  Newton  paid  $35  per  front  foot  for  Lot  7.  He 
had  a  6-ft.  cement  sidewalk  laid  in  front  of  the  lot,  cost- 
ing 12^  per  square  foot.  He  afterward  sold  the  lot  for 
$1500.     Did  he  gain  or  lose,  and  how  much  ? 


REVIEW  137 

REVIEW 
178.  Written  Exercises. 

1.  Add  34.125,  4.36,  180.006,  .67,  3.1416,  10.07, 

2.  Solve:  326.87-83.65;  9.82785  -  4.003  ;  346.85 - 

184. 

3.  Multiply:     32.064    by    .045  ;     $465.73    by    .08  ; 
$2456  by  .06. 

4.  Divide:  13.046  by  1.8;  143.78  by.06;  $120.78  by 
1.06. 

5.  Write  five  improper  fractions,  and  change  each  to 
a  whole  or  a  mixed  number. 

6.  Write  five  mixed  numbers,  and  change  each  to  an 
improper  fraction. 

7.  Write  five  proper  fractions  that  are  not  in  their 
lowest  terms,  and  change  each  to  lowest  terms. 

8.  Add  6f ,  4|,  71,  21,  14^3_. 

9.  From  28^\  subtract  9y\.     From  834f  take  186f 

10.  Multiply  8|  by  6| ;  683|  by  3| ;  31 1  by  18. 

11.  Divide  654|  by  9 ;  195J  by  7 ;  200  by  3J. 

12.  Divide  f  by  I ;  6f  by  71;  8|  by  7|. 

^     n         1       A    '      vf        12x14x21  X  9 

13.  Cancel  and  simplity :   -— — -- — -— — — -  • 

^    ^     16  X  20  X  15  X  30 

14.  Reduce  to  lowest  terms :  |^,  -5^,  ||f ,  l^\,  ^^* 

T.  A        4.    1         w  75     66^    331      40     12J 

15.  Reduce  to  lowest  terms:  — ,  ^,  ^,  — ,  ^, 

100'  100 ■ 


$8 

FRACTIONS 

179.  Oral  Exercises. 

Solve  : 

1.    12  xj 

10.   1  off  25 

19. 

f    of  21  da. 

2.    24  x| 

11.   1  of  140 

20. 

■^  of  84  yd. 

3.    15  X  J 

12.    1 

of  21  ft. 

21. 

1    of  48  mo 

4.     10  Xf 

13.    f 

of  64  mi. 

22. 

Jf  of  30  da. 

5.    18  x| 

14.    1 

of  36  in. 

23. 

f    of  120 

6.     14  Xf 

15.    1 

of  18  in. 

24. 

1    of  $120 

7.    30  x| 

16.    f 

of  35  yd. 

25. 

J    of  160 

8.    16  xf 

17.    f 

of  24  lb. 

26. 

f    of  170 

9.    27  X  1 

18.    1 

of  45  lb. 

27. 

f    of  $100 

180.  Oral  Exercises. 

Find  the  quotient  of : 

1.    1   ^4 

S. 

1^6 

9.    1,\ 

-r-4 

13.   2J-i-5 

2.    lf-^6 

6. 

1^10 

10.   6f 

^3 

14.    31-*.  8 

3.   {i^7 

7. 

1^2 

11.  m 

-^20 

15.    4j-*-4 

4.   if^5 

8. 

1^8 

12.    If 

-4-10 

16.    9f-i-4 

181.  Oral  Exercises. 

Find  the  product  of : 

• 

1.    ioii 

'•   f 

off 

13 

■   *xi 

2-  ioii 

8-    f 

of| 

14. 

|x| 

3.    foff 

9-    1 

off 

15. 

?xi 

4-    fofj 

10.     J,  Of  H 

16. 

|xf 

5.    -|0fA 

u.  1 

off 

17. 

*xi      . 

6.    fofi 

12.    1 

off 

18. 

JxH 

REVIEW  139 

182,  Oral  Exercises. 

1.  If  I  of  a  ton  of  coal  costs  $6,  what  is  the  cost  of  a 
ton? 

2.  If  f  of  the  cost  of  a  farm  is  f  2400,  what  is  the  cost 
of  the  farm  ?     What  is  |  of  the  cost  of  the  farm  ? 

3.  If  I  of  the  cost  of  a  carriage  is  $80,  what  is  the 
cost  of  the  carriage  ?  What  is  ^  of  the  cost  of  the 
carriage  ? 

4.  A  farmer  sold  -|  of  his  crop  of  oats  for  f  160.  At 
the  same  rate,  how  much  was  the  entire  crop  worth  ?  ^  of 
the  crop  ? 

5.  Some  men  entered  into  partnership.  One  man 
contributed  $800,  which  was  f  of  the  capital  invested. 
How  much  capital  was  invested  ?  How  much  was  con- 
tributed by  one  of  the  partners  who  furnished  J  of  the 
capital  ? 

6.  A  man  sold  |  of  his  land  for  f  1200.  At  this  rate, 
what  was  the  value  of  all  his  land  ? 

7.  A  poultry  dealer  sold  80  turkeys  and  then  had  ^  of 
his  stock  left.  What  part  of  his  stock  of  turkeys  did  he 
sell  ?     How  many  turkeys  had  he  at  first  ? 

8.  After  spending 'S 18  for  an  overcoat,  a  man  had  $6 
left.  What  part  of  his  money  did  he  spend  ?  What  part 
of  his  money  did  he  have  left  ? 

9.  After  traveling  24  miles,  a  man  still  had  f  of 
his  journey  to  travel.  Find  the  length  of  the  entire 
journey. 

10.  Mary  had  |  as  much  money  as  Ethel.  If  Mary  had 
60  ^,  how  much  did  Ethel  have  ? 


140  FRACTIONS 

11.  If  George  has  f  as  many  books  as  Walter,  and 
George  has  12  books,  how  many  books  has  Walter  ? 

12.  120  is  f  of ^  of  $20  = 

13.  $35  is  ^  of fof$35  = 

14.  After  increasing  his  farm  by  buying  |  as  many 
acres  as  his  farm  contained,  a  farmer  owned  120 
acres.  How  many  acres  did  he  own  before  making  the 
purchase  ? 

15.  Ethel  weighs  J  more  than  Edna.  Ethel's  weight  is 
105  lb.     What  is  Edna's  weight  ? 

16.  Thomas  solved  |  more  problems  than  Henry.  He 
solved  6  more  problems  than  Henry.  How  many  problems 
did  Henry  solve  ?     How  many  did  Thomas  solve  ? 

17.  $80  is  If  (f )  times  what  amount  ? 

18.  $120  is  2|  times  what  amount  ? 

19.  $200isfofic.     $60  is  IJ  of  a;. 

20.  What  amount  less  |  of  itself  equals  $100  ? 

21.  What  amount  less  |  of  itself  equals  $60  ? 

22.  $1200  is  2|  times  x,  ^  of  some  amount  is  $160. 
What  is  the  amount  ? 

23.  After  gaining  -J  of  his  capital,  a  merchant  had 
$14,000.     Find  the  amount  of  his  capital  at  first. 

24.  After  buying  3  books,  a  girl  had  8  books.  The 
number  of  books  bought  was  what  part  of  the  number  she 
previously  had  ? 

25.  $80  is f  of .   $120  is  J  of .   $90 is}  of . 

26.  $60  is  1 J  times $150  is  1 J  times . 

27.  $6 -4- $.75.     $9-<-$1.50.     $50 +  $1.25. 


flEVIEW  141 

183.   Oral  Exercises. 

1.  If  I  of  the  cost  of  a  pair  of  skates  is  60  ^,  the  cost  of 
the  pair  of  skates  is  how  many  times  60  ^  ? 

2.  If  J  of  the  cost  of  a  desk  is  $3,  the  cost  of  the  desk 
is  how  many  times  $3  ? 

3.  Compare  |  with  |.  Show  by  a  diagram  that  |  is 
IJ  times  |,  or  ^  times  |. 

4.  If  I  of  the  cost  of  a  table  is  $9,  the  cost  of  the  table 
is  how  many  times  $9  ? 

5.  I  of  the  cost  of  a  clock  is  $S,  In  finding  the  cost 
of  the  clock  we  may  find  ^  of  its  cost,  and  then  J  of  its 
cost.  Show  that  multiplying  $8  by  J  is  the  same  as  find- 
ing first  ^  of  the  cost,  and  then  f  of  its  cost. 

6.  If  I  of  the  value  of  a  horse  is  $60,  what  is  its  value? 

7.  If  30  sacks  of  oats  is  |  of  the  yield  per  acre,  what  is 
the  yield  per  acre  ?     Find  the  answer  in  two  ways. 

S.  A  man  sold  |  of  his  crop  of  apples  for  $120.  At 
the  same  rate,  what  was  the  value  of  his  entire  crop  ? 

9.  If  I  c'  a  ton  of  coal  costs  86,  what  is  the  cost  per 
ton  ?  Are  these  two  solutions  identical  in  character  ? 

A.     3)16,  'B.    $2 

f  2,  cost  of  {  ton.  ^^  >^  i  =  S8 

__4  ^ 

$8,  cost  of  1  ton.  1 

10.  If  I  of  the  cost  of  a  farm  is  f  6000,  what  is  the  cost 
of  the  farm  ?     What  is  ^  the  cost  of  the  farm  ? 

11.  After  selling  |  of  his  sheep,  a  farmer  had  60  sheep 
left.     How  many  had  he  at  first  ? 

12.  A  boy  sold  16  papers,  which  was  J  of  all  he  had. 
How  many  papers  had  he  at  first  ? 


142  FRACTIONS 

13.  After  solving  8  problems,  a  girl  had  |  of  her  prob- 
lems yet  to  solve.  How  many  problems  had  she  to  solve 
at  first  ? 

14.  By  selling  an  article  for  45^,  a  merchant  gained  J 
of  the  cost.     Find  the  cost  of  the  article. 

15.  By  selling  a  horse  for  '1^90,  a  man  lost  J^  of  its  cost. 
For  what  part  of  the  cost  did  he  sell  the  horse  ?  Find 
the  cost  of  the  horse. 

16.  By  selling  a  book  for  60^,  a  boy  lost  |  of  its  cost. 
For  what  part  of  its  cost  did  he  sell  the  book  ?  Find  the 
cost  of  the  book. 

17.  Two  boys  bought  a  sled  in  partnership,  one  paying 
I  of  its  cost  and  the  other  paying  |  of  its  cost.  The  boy 
who  paid  |  of  its  cost  paid  70^.  Find  the  cost  of  the 
sled. 

18.  After  having  his  salary  increased  by  J,  a  boy  re- 
ceived $)20  a  month.  What  was  his  salary  before  it  was 
raised  ? 

19.  A  dealer  advertised  second-hand  books  at  f  of 
their  ordinary  price.  At  what  price  does  he  sell  a  book 
that  costs  60^  when  new?  What  is  the  price  when  new 
of  a  book  which  he  sells  for  90^? 

184.     Written  Exercises. 

Find  the  whole  when  the  part  is  given  : 


1. 

112  is  1 

7. 

160  A.  is  { 

13. 

3|  mi.  is  J 

2. 

120  is  1 

8. 

320  rd.  is  | 

14. 

H  gal.  is  A 

3. 

75  mi.  is  | 

9. 

$42.50  is  f 

15. 

81  ft.  is  5^ 

4. 

lis} 

10. 

36  yd.  is  1 

16. 

$1.20  is  1 

5. 

fT.isf 

11. 

90  ft.  is  f 

17. 

3|isA 

6. 

$6400  is  f 

12. 

5280  is  1 

18. 

^44  is  f 

'  f 

of   the 

;  1 

of  the 

t¥o 

of  the 

REVIEW  143 

185.   Oral  Exercises. 

1.  Find  the  whole  amount  when  |  of  the  amount  is 

$60;  is  $18;  is  $54;  is  $90;  is  $240;  is  $1500. 

2.  Find  the  whole  amount  when  $  36  is  |  of  the 
amount;  |  of  the  amount;  -f^  of  the  amount;  -^^  of  the 
amount;  |  of  the  amount. 

3.  Find  the  whole  amount  when  $120  is  1^  times  the 
amount;  IJ  times  the  amount;  1^  times  the  amount;  IJ 
times  the  amount;  If  times  the  amount. 

4.  Find  the  whole  amount  when  $240  is 
amount;    f  of  the  amount;   |  of  the  amount 
amount;  |^  of  the  amount;  ^  of  the  amount. 

5.  Find  the  whole  amount  when  $600  is 
amount;  1|^  of  the  amount;  ^|gof  the  amount;  -^-^^  of  the 
amount;  -^^^  of  the  amount;  \^  of  the  amount. 

6.  A  boy  walked  2  blocks,  which  was  ^  of  the  distance 
from  his  home  to  the  schoolhouse.  How  many  blocks 
must  he  walk  in  going  to  and  coming  from  school  each 
day,  if  he  goes  home  for  lunch  ? 

7.  Charles  weighs  -^  less  than  Albert.  The  difference 
in  their  weight  is  11  lb.     How  much  does  each  weigh? 

8.  Margaret  wrote  7  more  words  than  Emma,  which 
was  ^  more  words  than  Emma  wrote.  How  many  words 
did  each  write  ?     How  many  did  both  together  write  ? 

9.  A  collector  charged  ^  of  the  amount  of  a  certain 
bill  for  collecting  it.  Find  the  amount  of  the  bill,  if  the 
creditor  received  $24. 

10.  After  selling  60  acres,  a  farmer  had  |  as  much  land 
left.     How  many  acres  had  he  before  making  the  sale  ? 

11.  $80  is  I  of .     90  mi.  is  f  of 


144  FRACTIONS 

REVIEW 

186.  Written  Exercises. 

1.  47f-14|  9.  87 -66f  17.  2|x3| 

2.  93|-52f  10.  47|+62f  is.  41- f 

3.  48fx84f  11.  19fx38|  19.  3f-2| 

4.  9|-4J  12.  2fx6f  20.  7f+8J 

5.  324|x|  13.  96|x7|  21.  |  =  2% 

6.  453|-h5  14.  30Jx45|  22.  87  =  f 

7.  526|^f  15.  897|4-6  23.  |x|xf 

8.  736fx5|  16.  7301  x  I  24.  3^x4| 

187.  Written  Exercises. 

1.  Find  the  value  of  |  of  a  farm  of  160  A.  at  f  85  per 
acre. 

2.  A  man  sold  |  of  his  farm  for  $4800.     At  the  same 
rate,  what  was  the  value  of  his  entire  farm? 

3.  If  3|  yd.  of  cloth  cost  #2.25,  what  is  the  cost  per 
yard? 

4.  Find  the  cost  of  8|  yd.  of  silk  at  $1.14  per  yard. 

5.  Find  the  cost  of  a  roast  of  lamb  weighing  4|  lb.  at 
IQj^  per  pound. 

6.  A  turkey   weighing  9^  lb.   was  bought  for  f  1.90. 
Find  the  price  paid  per  pound. 

7.  Express  in  cents  and  find  the  sum  of  the  following : 

^.  $|,  $i,  I  J,  ^.  li,  $1  $i,  H.  ^h  H^  ^h  H' 

8    Write  ten  improper  fractions  and  change  them  to 
whole  or  mixed  numbers. 

9.  Write  ten  mixed  numbers  and  change  them  to  im- 
proper fractions. 


NUMBER  RELATIONS  145 

NUMBER  RELATIONS 
188.  Oral  Exercises. 

Express  all  fractional  parts  in  their  lowest  terms. 

1.  What  part  of  10  is  5  ?  of  6  is  3  ?  of  8  is  2  ?  of  12  is 
4  ?  of  20  is  5  ?  of  30  is  6  ? 

2.  What  is  the  ratio  of  5  to  15?  of  6  to  12?  of  8  to 
24?  of  9  to  81?  of  T  to  56?  of  20  to  4?  of  28  to  7?  of  42 
to  6? 

3.  What  part  of  4  is  3  ?  of  8  is  5  ?  of  9  is  7  ?  of  11  is 
3?  of  10  is  6? 

4.  What  is  the  ratio  of  3  to  5  ?  of  5  to  3  ?  of  3  to  11  ? 
of  11  to  3  ?  of  7  to  9  ?  of  9  to  7  ? 

5.  What  is  the  ratio  of  6  sacks  of  oats  to  18  sacks  of 
oats  ?  6  T.  of  coal  will  cost  what  part  of  the  cost  of  18 
T.  ?     18  T.  will  cost  how  many  times  the  cost  of  6  T.  ? 

6.  If  5  sacks  of  flour  cost  $7.50,  how  much  will  10 
sacks  cost  ? 

7.  If  a  boy  earns  i3  in  4  da.,  how  much  will  he  earn 
in  16  da.  ? 

8.  If  a  boy  rides  at  the  rate  of  7  mi.  in  2  hr.,  how  far 
at  the  same  rate  will  he  ride  in  6  hr.  ? 

9.  If  12  pads  cost  1.60,  how  much  will  36  pads  cost? 

10.  If  3  T.  of  coal  cost  $24,  how  much  will  9  T.  cost  ? 

11.  What  number  expresses  the  ratio  of  4  lb.  to  8  lb.  ? 
of  5  lb.  to  20  lb.  ?  of  15  yd.  to  5  yd.?  of  20  A.  to  4 
A.?  of  $20  to  $30?  of  $24  to  $36?  of  18  bu.  to  24  bu.? 
of  36  ft.  to  21  ft.?  of  48  mi.  to  36  mi.?  of  $25  to  $50? 

12.  What  fraction  expresses  the  ratio  of  5^  to  25^?  of 
10^  to  50^?  of  20^  to  100^?  of  25^  to  100^?  of  25^  to 
75^?  of  10^  to  40^?  of  5^  to  45^?  of  20)^  to  50^? 

iK.  —  10 


146  J'ractionS 

13.  What  fraction  expresses  the  ratio  of  3  qt.  to  4  qt.? 
of  5  mi.  to  8  mi.?  of  4  lb.  to  6  lb.?  of  8  bu.  to  12  bu.?  of 
12  yd.  to  9  yd.  ?  of  20  mi.  to  15  mi.?  of  2b^  to  40^?  of  18 
yr.  to  12  yr.?  of  6  mo.  to  9  mo.?  of  9  mo.  to  12  mo.? 

14.  What  part  of  $1  is  25^?  50^?  75^?  40^?  60^? 
70^?  80^?  90^?  5/?  lOj^?  20/?  30/? 

15.  What  part  of  |1  is  12|/?  37|/?  621/?  87^/? 
33-1-/?  662^?  162/?  831/?  142/9  8J/? 

16.  What  part  of  1  ft.  is  2  in.?  3  in.  ?  4  in.  ?  5  in.  ?  6  in.  ? 

7  in.?  8  in.?  9  in.  ?  10  in.?  11  in.  ? 

17.  What  part  of  1  yd.  is  2  in.?  3  in.?  4  in.?  6  in.? 

8  in.?  9in.  ?  12  in.?   18  in.?   20  in.  ?  24  in.?  30  in.?   13 
in.  ?  21  in.  ? 

18.  Whatpartof  lib.  isJoz.?4oz.?  8oz.?  12  oz.?  7oz.? 

'     19.    What  part  of  1  yr.  is  2  mo.  ?  3  mo.  ?  4  mo.  ?  5  mo.  ? 
6  mo.?  7  mo.?  8  mo.?  9  mo.?  10  mo.?  11  mo.? 

20.  What  part  of  1  da.  is  2  hr.  ?  3  hr.  ?  4  hr.  ?  5  hr.  ? 
6  hr.?  8  hr.?  10  hr.  ?  12  hr.  ?  15  hr.?  16  hr.  ?  18  hr.? 
20  hr.? 

21.  What  part  of  1  hr.  is  5  min.  ?  10  min.  ?  15  min.  ? 
20  min.  ?  25  min.  ?  30  min.  ?  35  min.  ?  40  min.  ?  45  min.  ? 
50  min.  ?  ^^  min.  ?  17  min.  ? 

22.  What  part  of  1  mi.  is  10  rd.  ?  20  rd.  ?  40  rd.  ? 
80  rd.?  60  rd.?  90  rd.  ? 

23.  What  part  of  1  T.  is  1000  lb.  ?  500  lb.  ?  250  lb.  ? 
200  lb.  ?  100  lb.  ? 

24.  What  part  of  1  section  of  land  (1  sq.  mi.,  or  640  A.) 
is  320  A.?  160  A.?  80  A.  ?  40  A.?  20  A.? 


ALIQUOT  PARTS  147 

189.  Aliquot  Parts. 

1.  Name  several  amounts  that  are  exactly  contained  in 

136. 

2.  How  many  times  is  each  of  the  following  contained 
in$l:  50^?  25^?  12^^?  10)^?  20^?  5/?  4^?  2^? 

3.  A  part  of  a  number  or  a  quantity  that  will  divide 
it  without  a  remainder  is  called  an  aliquot  part.  Name 
several  aliquot  parts  of  100. 

4.  What  part  of  $1  is  each:  50/?  25^?  10^?  20)^? 
6/?  121/?  331^?  16|/?  Hi/?  6|/?  2/?  4/?  14^/? 

5.  If  40  sheep  can  be  bought  for  $100,  how  many  sheep 
can  be  bought  for  120  Q  of  |100)?  for  $25?  for  $12.50? 
for  110?  for  $5? 

6.  If  100  sacks  of  potatoes  cost  $80,  how  much  will  25 
sacks  cost?  50  sacks?  10  sacks?  5  sacks?  20  sacks? 

7.  How  much  will  30  yards  of  cloth  cost  at  $1  a  yard? 
at  25/  a  yard?  at  12^/  a  yard?  at  20/  a  yard?  at  16|/  a 
yard?  at  33J/  a  yard?  at  8^/  a  yard? 

8.  From  the  cost  of  any  number  of  articles  at  $1  each 
how  may  the  cost  of  the  same  number  of  articles  at  25/ 
each  be  found?  at  50/  each?  at  20/  each?  at  121/  each? 
at  33^/ each? 

190.  Memorize  the  following  fractional  parts  of  1 : 


.50=    i 

.20   =   i 

^^^  =  i 

.40   =1 

.25=    I 

.83J=   i 

.37^  =  1 

.60   =1 

•W=rV 

.75=   1 

.66|=   1 
.05   =^V 

•62J  =  f 
•87|=J 

•16f  =  J 
•14?  =  f 

From  the  above  table  construct   a  table  showing   the 
same  fractional  parts  of  $1 ;  of  100  ;  of  $100 ;  of  1000. 


1. 

At  50  ^  each. 

2. 

At  25  ^  each. 

3. 

At  75  ^  each. 

4. 

At  121^  each. 

5. 

At  331^  each. 

6. 

At  66|^each, 

148  FRACTIONS 

191.  Oral  Exercises. 

From  the  cost  of  120  articles  at  f  1  each  find  the  cost: 

7.  At  20  ^  each. 

8.  At37|^each. 

9.  At  621^  each. 

10.  At  40  ^  each. 

11.  At  60  ^  each. 

12.  At  80  /  each. 

192.  Written  Exercises. 

Solve  eacli  by  the  shortest  method. 

1.  Find  the  cost  of  24  yd.  of  cloth  at  37^^  per  yard. 
Suggestion  :    At  $1  per  yard  the  cloth  would  cost  $24. 

2.  Find  the  cost  of  24  yd.  of  cloth  at  87 J  i  per  yard. 
Suggestion  :    87^^  per  yard  is  \  less  than  $1  per  yard. 

3.  Find  the  cost  of  24  yd.  of  cloth  at  66|^  per  yard. 

4.  Find  the  cost  of  16  articles  at  1 25  each;  at  $250 
each  (1  of  $1000)  ;  at  $125  each  (^  of  $1000)  ;  at  $75  each 
(J  less  than  $100)  ;  at  $37.50  each. 

Suggestion  :    At  $100  each  the  16  articles  would  cost  %  1600. 

5.  How  many  articles  can  be  bought  for  $48  at  $1 
each?  at  25^  each?  at  331^  each?  at  66|j^  each?  at  Vl\^ 
each?  at  20)^  each?  at  37|  j^^  each? 

193.  Short  Methods. 

Solve  each,  using  the  shortest  method: 

1.  $40x25  5.  $2040x.l2J  9.  400  1b.  x. 625 

2.  $120x25  6.  640A.  x37J         lo.  $8.60x75 

3.  $80x250  7.  240  mi.  X. 125       ii.  $5.60x750 

4.  60mi.  x33i         8.  36  ft.  x  125  12.  $4.64x12.5 


ALIQUOT  PARTS  149 

194.    1.    Divide  by  25:  12;  $16;  640  A. 

To  divide  hy  25,  divide  hy  100  and  multiply  the  quotient 
by  4:, 

2.  State  a  short  method  of  dividing  a  number  by  250; 
by  50  ;  by  SSJ ;  by  66| ;  by  371 ;  by  12| ;  by  375 ;  by  75 ; 
by  125;  by  .25;  by  .125;  by  12.5;  by  2.5;  by  62.5;  by 
625 ;  by  500. 


ivide  : 

3.    $400  by  25 

10. 

2240  lb.  by  .25 

4.   $  300  by  250 

11. 

2000  lb.  by  2.5 

5.    $600  by  50 

12. 

5280  ft.  by  37.5  ft. 

6.    320  rd.  by  .125 

13. 

1728  by  250 

7.    640  mi.  by  12.5 

14. 

$400  by  87.5 

8.    540  ft.  by  331 

15. 

$  3200  by  f  625 

9.    120  yr.  by  66f 

16. 

$1500  by  12.50 

195.  1.  What  is  the  cost  of  24  yd.  of  cloth  at  50  /  per 
yard?  at  12|  /  per  yard?  at  16|)^  per  yard?  at  75/  per 
yard  ?  at  87^  ^  per  yard  ?  at  37|  /  per  yard  ? 

2.  How  many  yards  of  cloth  can  be  purchased  for  $12 
at  25  /  a  yard  ?  at  12|  /  a  yard  ?  at  61  /  a  yard  ?  at  37^  / 
a  yard  ?  at  33^  /  a  yard  ?  a\;  66 J  /  a  yard  ?  at  $1.50  a  yard? 
at  $1,331  a  yard?  at  $1.20  a  yard? 

3.  If  40  acres  of  land  cost  $  2000,  how  much  will  50 
acres  cost  at  the  same  rate  ?  60  acres  ?  100  acres  ?  45  acres  ? 
55  acres  (40  acres  +  |  of  40  acres  -4-  J  of  -J  of  40  acres)  ? 

4.  George  White  paid  Thomas  Evans  $  12  for  the  loan 
of  some  money  for  60  da.  At  the  same  rate,  how  much 
must  he  pay  for  the  use  of  the  same  sum  for  90  da.  ?  for 
30  da.  ?  for  75  da.  (60  da.  +  I  of  60  da.)  ?  for  120  da.  ? 
for  70  da.?  for  50  da.  ?  for  20  da.  ?  for  80  da.  ? 


150  FRACTIONS 

REVIEW 

196.  1.  Draw  a  square  and  show  the  following  parts 
of  it  :  .50,  .25,  .75,  .121  .37^,  .871,  .33J,  .66|. 

2.  Draw  a  square,  and  divide  it  into  as  many  equal 
parts  as  are  necessary  to  show  either  J  or  J  of  the  square. 

3.  What  is  the  1.  c.  m.  of  2,  3,  9?   of  2,  3,  4,  6? 

4.  Draw  a  square,  and  divide  it  into  as  many  equal 
parts  as  are  necessary  to  show  all  of  the  following  parts : 
h  h  h  h  h     S^C)w  on  the  square  the  parts  |,  |,  |,  |. 

5.  Describe  a  cubic  foot.  Think  of  a  box  whose  inside 
measure  is  1  ft.  by  1  ft.  by  1  ft.  How  many  bricks  do 
you  think  the  box  will  contain?  How  can  you  find  the 
exact  number  it  will  contain?  If  this  box  is  watertight, 
how  many  gallons  will  it  contain?     (1  gal.  =  231  cu.  in.) 

6.  A  boy  made  a  bookcase.  The  top  of  the  lower  shelf 
is  Q"  from  the  floor.  The  space  between  the  lower  shelf 
and  the  top  of  the  case  is  3'  6''.  The  case  contains  four 
shelves.  The  space  between  the  two  lower  shelves  is  12''. 
The  other  shelves  are  placed  so  that  the  distance  between 
them  is  exactly  equal.  How  far  apart  are  they  if  the 
shelves  are  each  ^"  in  thickness?  The  width  of  the  case 
is  2'  6",  and  the  depth  1'.  Draw  a  diagram,  using  the 
scale  2'' =  1'. 

197.  1.    Divide  each  by  10  :  47;  138.40;  f.80;  3.1416. 

2.  Divide  by  100:  22001b. ;  5280  ft.;  1760  yd.;  1457.50. 

3.  State  a  short  method  for  multiplying  by  10 ;  by 
100;  by  25;  by  33^;  by  66|;  byT5;  by  12|;  by  .25; 
by  .87J;  by  37|.     Give  several  illustrations  of  each. 

4.  Explain  what  is  meant  by  the  reciprocal  of  any 
number.    What  is  the  reciprocal  of  |?  of  8?  of  ^?  of  5^? 


REVIEW  161 

5.  Divide  f  ft.  by  ^\  ft.;  6.2  by  .02;  if  by  6 ;  if 
by  3. 

6.  State  how  to  multiply  and  how  to  divide  a  fraction 
by  a  fraction  ;  a  whole  number  by  a  fraction  ;  a  fraction 
by  a  whole  number;  a  mixed  number  by  a  whole  number 
or  a  mixed  number.     Illustrate  each. 

7.  State  how  to  determine  the  place  of  the  decimal 
point  in  the  quotient  in  division  of  decimals.  Fix  the 
decimal  point  in:    .002)4.8368;  2.36)13:4;  34)4.275. 

8.  The  question,  How  many  square  inches  are  there 
in  1  square  foot?  is  answered  by  the  number  144.  Ask 
a  similar  question  that  is  answered  by  each  of  the  follow- 
ing: 3,  12,  9,  4,  2,  60,  24,  51  144,  7,  320,  231,  1728,  5280, 
365, 640, 27, 52, 160, 128,  2150.42,  8,  32, 100, 16,  2000,  2240. 

9.  The  question.  What  is  ^  of  100?  is  answered  by 
the  number  121.  Ask  a  question  concerning  a  fractional 
part  of  100  that  is  answered  by  each  of  the  following :  20, 
75,  25,  121,  871,  5,  621    qqi^  871    331,  40,  10,  80,  16|. 

10.  The  question.  What  is  the  ratio  of  10^^  to  |1?  is 
answered  by  the  number  -^q.  Ask  a  question  concerning 
the  ratio  of  some  quantity  to  $1  that  is  answered  by 
each-  1  -^  4  4  1  4  ^  X  4  i   ^  J-  -1-  4  ^  4^   2  -5^  4 

Ctn^u  .    2^  3?  ^1  ^5  6'  1'  5'   8'  ^'  ^'   5'   10'  2  0'  1'  2'  4'  "^^  3' F* 

198,  The  first  number  is  the  product  of  two  numbers. 
The  second  number  is  one  factor.     Find  the  other  factor. 


1. 

36,  6 

7. 

1.5,  .5 

13. 

816.40,  1 

2. 

30,1 

8. 

$.15,  3 

14. 

$20,  .331 

3. 

15,  .5 

9. 

1.15,  5^ 

15. 

96  ft.,  .121 

4. 

l'2 

10. 

11.80,  .06 

16. 

96  ft.,  1 

5. 

f'i 

11. 

$30,  .05 

17. 

$34.40,  .08 

6. 

.5,10 

12. 

125.60,  .08 

18. 

$10,  .04 

152 


FRACTIONS 


199.  Oral  Exercises. 

Fill  in  the  amounts  omitted  under  each  heading 


Cost 

Selling  Price 

Gain 

Loss 

Part  Gained 

Part  Lost 

1.    140 

150 

$10 

i 

2.    125 

130 

X 

X 

3.   160 

$50 

X 

X 

4.    $80 

X 

$20 

X 

5.         X 

$100 

$25 

X 

6.        X 

$150 

$50 

X 

7.    $75 

X 

$50 

X 

8.      X 

$75 

X 

\ 

9.  $150 

X 

X 

\ 

10           X 

$110 

X 

tV 

11.  Express  as  hundredths  the  part  gained  or  lost  in 
each  of  the  above  exercises,  as  \  gain=  .25  gain. 

12.  Write  and  solve  ten  exercises  similar  to  Exs.  1-10 
above. 

13.  Example  5  above  may  be  stated  as  a  problem: 

Illustration:   A  man  sold  a  horse  for  $100  at  a  gain  of  $25. 
Find  the  cost  of  the  horse  and  what  part  the  gain  is  of  the  cost. 

14.  State  Exs.  1-4  and  6-10  as  problems. 

200.   1.    If  .04  of  some  amount  is  $10,  what  is  the 

amount? 

2.  By  what  must  $240  be  multiplied  to  produce  $12? 

3.  Multiply:  $600  by  .06;  $300  by  .04;  $80.50  by  .07. 

4.  Find  .06  of  $360;  of  $4  ;  of  $24;  of  $30;  of  80  mi. 

5.  If  4  times  some  amount  is  $16,  what  is  the  amount? 


REVIEW  153 

201.  Oral  Exercises. 

1.  If  12  articles  cost  $  36,  how  may  the  cost  of  6  articles 
be  found  without  finding  the  cost  of  1  article  ? 

2.  What  part  of  the  cost  of  12  articles  must  be  added 
to  the  cost  of  12  articles  to  give  the  cost  of  18  articles  ? 
of  15  articles  ?  of  14  articles  ?  of  16  articles  ?     Illustrate. 

3.  What  part  of  the  cost  of  6  articles  must  be  added  to 
the  cost  of  6  articles  to  give  the  cost  of  9  articles  ?  of  8 
articles  ?  of  7  articles  ?     Illustrate  each. 

4.  What  part  of  the  cost  of  6  articles  must  be  sub- 
tracted from  the  cost  of  6  articles  to  give  the  cost  of  5 
articles  ?  of  4  articles  ?     Illustrate  each. 

5.  When  the  cost  of  6  articles  is  known,  how  may  the 
cost  of  3  articles  be  found  ?  of  2  articles  ?  of  1  article  ?  of 
12  articles  ?     Illustrate  each. 


6.    When  the  cost  of  50  articles  is  known,  how  may  the 
►st  of  12J  articles  be 
articles  ?  of  75  articles  ? 


cost  of  12J  articles  be  found  ?  of   37J  articles  ?  of  62^ 


202.  1.  If  .06  times  some  amount  is  $12,  what  is 
the  amount?  If  .04  of  an  amount  is  $20,  what  is  the 
amount  ?     If  .03  of  an  amount  is  $  24,  what  is  the  amount  ? 

2.  145  is  .09  of  what  amount?  175  is  .15  of  what 
amount  ?     f  1.60  is  .08  of  what  amount? 

3.  $  15  is  1.25  of  what  amount  ?  is  .20  of  what  amount  ? 

4.  How  much  is  1.75  of  $80?  of  $200?  of  640 A.? 

5.  $  40  is  what  part  of  $  80  ?     J  =  yf  ^. 

6.  $20  is  how  many  hundredths  of  $40?  of  $80? 

7.  .6  of  600  is  .12  of  what  number? 

8.  .9  of  800  is  .3  of  what  number  ? 


154  FRACTIONS 

203.   Written  Exercises. 

Keep  each  result  until  all  the  problems  have  been  solved. 

1.  A  farmer  rented  a  field  60  rd.  long  and  40  rd. 
wide.     Find  the  number  of  acres  in  the  field. 

2.  The  annual  rent  of  the  field  was  $8.75  per  acre. 
Find  the  rent  of  the  field  for  1  yr. 

3.  The  farmer  planted  the  field  in  broom  corn,  which 
yielded  J  T.  to  the  acre.  Find  the  total  yield  of  broom 
corn.  Each  acre  of  broom  corn  yielded  1  T.  of  seed, 
valued  at  $  16  per  ton.     Find  the  value  of  the  seed. 

4.  The  farmer  sold  the  broom  corn  at  $80  a  ton. 
Find  the  value  of  the  crop. 

5.  The  farmer  paid  a  commission  merchant  $4  a 
ton  for  selling  the  broom  corn.     Find  the  commission. 

6.  The  commission  merchant  paid  $2.50  per  ton 
freight  and  f  .75  per  ton  cartage  on  the  broom  corn. 
How  much  should  he  remit  to  the  farmer,  after  deduct- 
ing these  expenses  and  his  commission  ? 

7.  The  expense  of  seed  and  of  planting  and  harvest^ 
ing  the  crop  amounted  to  $  15  per  acre.  How  much  was 
the  farmer's  net  profit  per  acre  from  the  crop  ? 

8.  If  25  lb.  of  broom  corn  are  used  in  making  1  doz. 
brooms,  how  many  dozen  brooms  can  be  made  from  the 
yield  of  1  A.? 

9.  If  the  manufacturer  sells  the  brooms  for  $2.50  per 
dozen,  how  much  does  he  receive  for  the  brooms  made 
from  the  yield  of  1  A.? 

10.    How  much  did  the  broom  corn  cost  per  pound  at 
$80  per  ton? 

'■^'^  11.    How  much  is  the  cost  of  the  broom  corn  used  in 
making  1  doz.  brooms  ? 


REVIEW  155 

12.  If  the  cost  of  labor  and  of  the  material  other  than 
the  broom  corn  is  f  .80  for  each  dozen  brooms,  how  much 
do  the  brooms  cost  the  manufacturer  per  dozen  ? 

13.  How  much  is  the  manufacturer's  profit  on  each 
dozen  brooms  ?  The  manufacturer's  profit  on  each  dozen 
brooms  is  what  part  of  the  cost  of  a  dozen  ? 

14.  A  wholesale  merchant  bought'the  brooms  from  the 
manufacturer  at  f  2.50  a  dozen  and  sold  them  to  retail 
dealers  at  a  profit  of  -^^  of  the  cost.  What  was  the  price 
of  the  brooms  per  dozen  to  the  retail  dealer? 

15.  The  retail  dealer  sold  the  brooms  to  his  customers 
at  a  profit  of  ^  of  the  cost  to  him.  Find  the  price  paid 
to  the  retail  dealer  for  each  broom. 

16."  If  a  retail  dealer's  net  profit  on  each  dozen  brooms 
is  ^  of  the  gross  profit,  how  much  was  his  net  profit  on  the 
sale  of  1  doz.  brooms  ? 

17.  Find  the  difference  between  the  cost  of  1  doz. 
brooms  to  the  manufacturer  and  the  cost  to  the  consumer. 

204.  Written  Exercises. 

1.  .06  of  some  amount  is  f  30.  Find  .03  of  the  same 
amount. 

Suggestion  :   .03  is  one  half  of  .06.     Take  one  half  of  $  30. 

2.  .06  of  an  amount  is  $30.     Find  .09  of  the  amount. 
Suggestion:    .09  is  one  half  more  than  .06. 

Follow  the  above  suggestions  in  the  solution  of  each : 

3.  Find  .06  of  16400.  From  the  answer  find  .02  of 
16400;  .03of|6400;  .09of  16400;  .08  (1  more  than. 06) 
of  16400 ;  .04  (i  less  than  .06)  of  16400 ;  .05  (J  less  than 
.06)  of  $6400;  .07  of  $6400,-   J.2  of  ^6400. 


156  FRACTIONS 

205.   Changing  Decimal  Fractions  to  Common  Fractions. 

1.  What  is  the  numerator  in  .375?  What  is  the 
denominator? 

2.  How  many  decimal  places  are  there  in  .375?  Write 
.375  as  a  common  fraction.  Compare  the  number  of 
decimal  places  in  .375  with  the  number  of  O's  in  -f^. 

3.  Change  .875  to  a  common  fraction. 

Model:   .875  =  ^V^  =  f 
To  change  a  decimal  fraction  to  a  common  fraction^  write 
the  numerator  of  the  fraction  over  the  denominator  of  the 
fraction.     Reduce  to  lowest  terms. 

Change  to  common  fractions : 

4.  .1,  .2,  .3,  .4,  .5,  .6,  .7,  .8,  .9. 

5.  .10,  .20,  .30,  .40,  .50,  .60,  .70,  .80,  .90. 

6.  .12,  .15,  .25,  .35,  .45,  .55,  Sb,  .75,  .85,  .95. 

7.  .125,'  .375,  .625,  .875,   .025,  .075,  .04,  .05,  .02. 

8.  Reduce  to  mixed  numbers  :  4.25,  26.5,  8.75,  15.375, 
45.125,  7.875,  12.625,  8.20,  35.60,  2.04,  5.40. 

9.  In  the  following  the  cents  and  mills  are  expressed 
decimally  as  fractions  of  a  dollar.  Write  with  the  cents 
expressed  as  common  fractions;  thus,  |56.40  =  'f  6|:  $5.25, 
$8.20,  $4.10,  $3.50,  $7.80,  $2.75,  $6.60,  $4,125,  $9,625, 
$8,375,  $4,875,  $25.30,  $15.05,  $4.01,  $7.90. 

10.  What  decimal  is  equivalent  to  J?   i?  |?  i?  |?  |? 

F  V  f?  I?  1^?  A?  T%? 

11.  Express  each  as  an  improper  fraction:  1.25,  1.10, 
1.20,  1.125,  2.40,  1.375,  1.625,  1.875,  3.6,  1.80. 

12.  Add  as  decimals:   .25,  .125,  .4,  .875,  .2,4.75,6.07. 
Change  to  common  fractions  and  add. 


Common  to  DfeciMAt  157 

S06.  Changing  Common  Fractions  to  Decimal  Fractions. 

1.  Change  |  to  a  decimal  fraction. 

3  -7-  4  is  the  same  as   -I-     A   fraction   is   an   indicated 

division,  in  which  the  numerator  is  the  dividend  and  the 

denominator  is  the  divisor.     The  division  indicated  by  | 

may  be  performed  by  placing  the  decimal  point  after   3 

and  dividing,  thus : 

.75 
I  =   .     '     .      The  fraction  |  has  been  reduced  to  a  decimal  fraction. 

2.  Perform  the  indicated  division.  Continue  the 
division  until  there  is  no  longer  a  remainder.     I,  |,  |^,  ^, 

h    h  h  h  h 

To  change  a  common  fraction  to  a  decimal  fraction^ 
divide  the  numerator  hy  the  denominator. 

3.  The  prime  factors  of  10  are  2  and  5.  Name  all  the 
numbers  to  30  which  have  no  other  prime  factors  than  2 
or  5.  Find  by  trial  whether  any  fraction  whose  denom- 
inator has  any  prime  factors  other  than  2  or  5  can  be 
changed  to  an  exact  decimal.  Which  of  the  fractions  in 
Sec.  207  can  be  reduced  to  exact  decimals  ? 

207.   Written  Exercises. 

Change  the  following  to  decimals.  Where  the  decimal 
is  inexact,  continue  the  division  to  three  places. 


1. 

f 

7. 

1 

13. 

A 

19. 

2| 

2S. 

$H 

2. 

f 

8. 

f 

14. 

^ 

20. 

m 

26. 

|7t 

3. 

\ 

9. 

1 

6 

15. 

f 

21. 

Hs 

27. 

»8fV 

4. 

i 

10. 

\ 

16. 

1 

22. 

H 

28. 

I7f 

5. 

f 

11. 

f 

17. 

i 

23. 

82V 

29. 

f5i 

6. 

i 

12. 

1 

18. 

A 

24. 

^2\ 

30. 

i6T-L 

158  FRACTIONS 

208.  Changing  Fractions  to  Hundredths. 

1.  Change  |  to  a  fraction  whose  denominator  is  100. 

4        X 
Model:        r— ^j — 7"         ^   i^  contained  in   100  twenty   times. 

^      1^^        Multiply  both  terms  of  i  by  20.     i  =  ^%. 
Since  1  is  equivalent  to  {^^,  ^  is  equivalent  to  ^q%,  and  ^  is  equiva- 
lent to  j%. 

2.  Change   to   fractions  whose  denominators  are  100 : 

1'  h  h  h  h  i  ft,'  sV-  #5'  h  h  h  h  h  h  h  h  T^- 

3.  Express  as  decimal  fractions  each  of  the  fractions  in 
Ex.  2. 

4.  Memorize  the  following  : 

1  =  ?^  =  .334 
8  100    ^ 

3  100    ^ 

8   100    ' 

?  =  ?li  =  .37i 
8   100    ^ 

'^  =  ^^  =  .621 
8  100     ^ 

8   100    * 

6  100    ^ 

l  =  lil  =  .14? 

7  100    ^ 

5.  Express  as  common  fractions  in  lowest  terms:    .25, 
.20,  .40,  .50,  .60,  .70,  .75,  .80,  .05,  .02,  .04,  .10,  .90,  .01. 

6.  Write  with  the  fractional  part  expressed  as  a  deci- 
mal  :  7^,  4f ,  6f ,  8J,  9|,  4^,  Z^,  S^,  12^,  9f 

7.  Express  as  dollars  and   cents  and   add     t8J,  i3|, 
$9J,  f4fj,  $7^,  *8^,  il2|.  |15f,  IWjV,  i72i5. 


1  = 

.ioo_ 

100 

1 

1_ 

2 

60 
100 

.50 

1_ 
4 

25 

'ioo~ 

.25 

3_ 
4 

75 
100 

.75 

1_ 
5 

20 
100 

.20 

2  = 
6 

.  40  _ 
100 

.40 

3_ 
5 

60 
100 

.60 

4_ 
5 

80  _ 
100 

.80 

1  _ 

20" 

5 
100 

.05 

1  _ 

25 

4 
100 

.04 

1  _ 

50 

2 
100 

.02 

1  _ 
12 

100 

.08^ 

1 
10 

_  10  _ 
"100 

.10 

3  _ 
10 

-  30  _ 
"100 

.30 

7 
10 

70  _ 
"100 

.70 

9  _ 
10" 

.  00  _ 
"100 

.90 

REVIEW  159 

209.  Written  Exercises. 

1.  There  are  2000  lb.  in  a  ton.  How  many  tons  of 
hay  are  there  in  5400  lb.  ? 

2.  At  $8.50  per  ton,  how  much  will  6500  lb.  of  hay 
cost? 

3.  At  f  10.50  per  ton,  how  much  will  950  lb.  of  coal 
cost? 

4.  How  many  hundredweight  (100  lb.)  are  575  lb.  ? 
At  $5.60  per  hundredweight,  how  much  will  a  farmer 
receive  for  some  hogs  weighing  3750  lb.  ? 

5.  At  $37.50  per  ton,  how  much  will  a  farmer  receive 
for  12,400  lb.  of  wheat  ? 

6.  At  $1.25  each,  how  many  hats  can  be  bought  for 

$20? 

7.  How  much  will  45.75  A.  of  land  cost  at  $65.50  per 
acre  ? 

8.  The  circumference  of  a  circle  is  3.1416  times  its 
diameter.  Find  the  diameter  of  a  tree,  the  circumference 
of  which  is  7.75  ft. 

9.  Find  the  circumference  of  a  cylindrical  tank,  the 
diameter  of  which  is  4  ft.  9  in.  (4.75). 

10.  At  $38.50  each,  how  much  will  14  cows  cost  ? 

11.  The  area  of  a  rectangle  is  42.625  sq.  in.     Its  length 
is  7.75  in.     How  wide  is  the  rectangle  ? 

12.  If  a  train  travels  at  an  average  rate  of  46.75  mi. 
per  hour,  in  how  many  hours  will  it  travel  390.6  mi.  ? 

13.  At  5J^  ($.0525)  per  pound,  how  many  pounds  of 
sugar  can  be  bought  for  $4.20? 

14.  When  hay  is  worth  $7.50  per  ton,  how  many  tons 
can  be  bought  for  $90? 


160  FRACTIONS 


LUMBER  MEASURE 


210.  1.  The  unit  used  in  measuring  lumber  is  the 
board  foot,  which  is  the  equivalent  of  a  piece  of  board  1  ft. 
long,  1  ft.  wide,  and  1  in.  thick. 

A  board  12  ft.  long,  12  in.  (1  ft.)  wide,  and  1  in.  or  less  in  thick- 
ness contains  12  times  1  board  foot,  or  12  board  feet  In  measuring 
lumber,  boards  less  than  1  in.  thick  are  considered  inch  boards.  The 
name  board  foot  is  generally  shortened  to  "  foot."  The  Roman 
jiumeral  "M"  is  used  to  denote  a  thousand  feet. 

2.  How  many  board  feet  are  there  in  a  piece  of  board 
1  ft.  long,  12  in.  (1  ft.)  wide,  and  2  in.  thick?  1  ft.  long, 
6  in.  (^  ft.)  wide,  and  2  in.  thick  ? 

3.  What  part  of  a  board  foot  is  there  in  a  piece  of  board 
1  ft.  long,  6  in.  Q  ft.)  wide,  and  1  in.  thick  ?  1  ft.  long, 
8  in.  (I  ft.)  wide,  and  1  in.  thick?  1  ft.  long,  4  in. 
(1^  ft.)  wide,  and  1  in.  thick? 

4.  A  piece  of  board  1  ft.  long,  6  in.  wide,  and  2  in.  thick 
contains  2  times  |  board  foot,  or  1  board  foot.  Explain. 
How  many  board  feet  are  there  in  a  piece  of  board  1  ft. 
long,  6  in.  wide,  and  3  in.  thick  ? 

To  find  the  number  of  hoard  feet  in  a  piece  of  lumber^ 
multiply  the  number  of  board  feet  in  one  foot  of  the  length  by 
the  number  of  feet  in  the  length  of  the  piece, 

5.  Find  the  number  of  board  feet  in  16  pieces  of  3"  by 

4^',  each  12  ft.  long. 

Model-  ^^^^  either  3"   or  4"  as 

the  width.    Taking  4"  as  the 

16  X  ?;2  X  3  X  4  board  feet  =     f^^.'  *''t"T'f '  f  ^"^ 

^^  yp  feet  m  1  ft.  of  the  length  is 

192  board  feet.  ^  ^  i^  ^^^^^  ^^®*5   and  in  1 
piece  12  ft.  long,   12  x  3  x  ^^ 
board  feet ;    and   in   16  pieoes,  16xl2x3x^  board  feet,  or  192 
board  feet. 


LUMBER  MEASURE  161 

211.  Oral  Exercises. 

Find  the  number  of  feet  in  a  piece  of  lumber  of  the 
following  dimensions: 

1.  V^  X  12'',  10  ft.  long  5.    4"  X    4'^  12  ft.  long 

2.  2'^  X  12'^    1  ft.  long  6.    4'^  X    8'',  15  ft.  long 

3.  2"  X    6",  14  ft.  long  7.    6'^  x    6'',  18  ft.  long 

4.  2'^  X    4'^  16  ft.  long  8.    1'^  X  16'^  15  ft.  long 

212.  Written  Exercises. 

1.  Find  the  number  of  feet  in  120  pieces  of  lumber,  each 
2"  by  ^"  by  16'. 

2.  Measure  various  pieces  of  lumber. 

3.  Find  the  cost  of  lumber  for  a  bridge  10  ft.  long,  if 
planks  3''  x  12''  x  14'  are  laid  over  four  timbers  8"  x  8" 

xl4'.     Boards  costing  118  per  M;  timbers  1 20  per  M. 

4.  Find  the  cost  of  the  lumber  for  a  5-board  fence 
around  an  orchard  160  ft.  by  240  ft.  The  boards  used 
are  6  in.  by  1  in.  by  16  ft.  and  cost  1 14  per  M.  The 
posts,  8  ft.  in  length,  are  set  8  ft.  apart,  and  are  made  of 
pieces  4  in.  by  4  in.  by  16  ft.,  costing  f  16  per  M. 

213.  Flooring. 

1.  When  tongued  and  grooved^  a  board  3  in.  wide  is 
2^  in.  wide  when  laid.  The  part  of  the  board  thus  lost  is 
^  of  the  width  covered  by  the  board  after  it  has  been  fitted. 
Explain.  If  168  ft.  of  flooring  3  in.  wide  are  needed  for  a 
certain  floor,  ^  as  much  must  be  added  if  tongued  and 
grooved  flooring  3  in.  wide  is  used.     Why? 

2.  Find  the  number  of  feet  of  flooring  needed  for  a 
room  24  ft.  wide  and  30  ft.  long,  if  tongued  and  grooved 
flooring  3  in.  wide,  |  in.  thick,  and  12  ft.  long  is  used. 
What  is  the  cost  of  the  flooring  at  $40  per  IVf" 

di)..   XB.  — 11 


162  FRACTIONS 

3.  What  part  of  5|-  in.  is  J  in.?  of  2J-  in,  is  J  in.? 
Having  found  the  number  of  feet  of  lumber  needed  to 
floor  a  certain  room  with  boards  6  in.  wide,  how  may 
the  number  of  feet  needed  to  floor  the  same  room  with 
boards  6  in.  wide  that  have  been  tongued  and  grooved 
be  found  ? 

214.   Shingling. 

1.  The  unit  of  shingling  is  a  square,  which  is  an  area 
of  100  square  feet. 

2.  When  shingles  have  been  laid,  about  4  inches  of 
their  length  is  exposed  to  the  weather.  The  average 
width  of  a  shingle  is  about  4  inches.  Consequently  the 
exposed  surface  of  one  shingle  is  about  16  square  inches, 
or  about  J  square  foot.  It  will  thus  take  about  900  shingles 
to  cover  a  square.  Allowing  for  waste,  1000  shingles  are 
estimated  for  a  square.  A  hunch  of  shingles  contains  250 
shingles.  How  many  bunches  should  be  allowed  to  each 
square  ? 

3.  Find  the  cost  of  the  shingles  necessary  to  cover  both 
sides  of  a  roof,  if  each  side  is  24'  by  40',  at  f  2.25  per 
thousand  shingles. 

Model  :  24  x  40  x  2  x  .01  x  1 2.25  =  x. 

The  number  of  square  feet  in  both  sides  of  the  roof  is  24  x  40  x  2, 
and  the  number  of  squares  is  .01  times  this  product.  The  cost  of  the 
shingles  is  ^2.25  multiplied  by  the  number  of  squares.     Why? 

4.  Find  the  cost  of  the  shingles  necessary  to  cover  both 
sides  of  a  roof,  if  each  side  is  36'  by  48',  and  the  shingles 
cost  $2.50  a  thousand. 

5.  Estimate  the  cost  of  shingles  to  cover  the  roof  of 
your  schoolhouse,  at  $2  a  thousand. 


yr. 
1907 

mo. 
1 

da. 

4 

1893 

5 

26 

DIFFERENCE   BETWEEN  DATES  163 

215.  Difference  between  Dates. 

1.  Walter  Harris  was  born  May  26,  1893.  How  old 
was  he  on  January  4,  1907? 

Model  :  Write  the  later  date  as  the  minuend  and 

the  earlier  date  as  the  subtrahend.  It  is 
evident  that  some  number  of  days  added  to 
26  da.  equal  1  mo.  and  4  da.  Subtract 
thus:  26  da.  and  4  da.  are  1  mo.  4  da. 
13  7  8      and  4   da.    (in   the    minuend)   are   8   da. 

Carry  1  mo.  to  5  mo.,  as  in  subtraction  of  integers.  6  mo.  and  6  mo. 
are  12  mo. ;  6  mo.  and  1  mo.  are  7  mo.  Carry  1  yr.  tp  1893.  Com- 
plete the  subtraction. 

2.  Find  your  age  by  subtraction. 

3.  Find  the  time  from  the  landing  of  Columbus  in 
America  to  the  date  when  the  Declaration  of  Independence 
was  signed. 

4.  Frank  Thomas  borrowed  1 750  of  Charles  Gray  on 
Oct.  8,  1902,  and  paid  it  on  July  2,  1903.  How  long  did 
he  have  the  money? 

5.  When  the  exact  number  of  days  between  two  dates 
that  are  less  than  a  year  apart  is  required,  it  is  necessary 
to  take  account  of  the  number  of  days  in  each  month  in- 
cluded, as  in  the  following :  Find  the  exact  number  of 
days  from  Jan.  4,  1907  to  April  11,  1907.  There  are  27 
full  days  left  in  January,  28  days  in  February,  31  days  in 
March,  and  11  days  in  April  (including  April  11),  or 
27  da.  +  28  da.  +  31  da.  + 11  da.,  or  97  da. 

6.  Find  the  exact  number  of  days  from  the  Fourth 
of  July  to  Christmas;  from  Christmas  to  May  1. 

7.  Mr.  Jenkins  borrowed  a  team  of  Mr.  Slate  on 
Aug.  21  and  returned  it  on  Nov.  15.  At  il.50  a  day, 
how  much  did  he  owe  for  the  use  of  the  team? 


164 


FRACTIONS 


o 


1} 


216.   Review  Exercises. 

1.  Make  a  drawing  to  represent  a  city  lot  40  ft.  front 
and  120  ft.  deep,  using  the  scale  1  in.  =  20  ft.  Using  the 
same  scale,  represent  at  the  back  of  the  lot  the  space  occu- 
pied by  a  barn  20  ft.  by  30  ft. 

2.  A  man  bought  a  tract  of  land  160  rd.  long  and  SOi^ 

rd.  wide.    How  many '  "^ 
acres  did  it  contain  ? 
The  tract  was  di- 
vided as  shown  in  the 

^  , figure.    Find  the  area 

of  each  field.  Find 
the  cost  of  fencing  the 
tract  as  shown  in  the 


/60rd. 


40ra/. 

40rcf. 

eOrd. 

1 

40rd. 

40rd 

1 

\ 

aord. 


4/t 


figure  at  f  1.25  per  rod. 

3.  A  field  containing  20  A.  is  40  rd.  wide.     How  long 

4.  Mr.  James  bought  Lot  2  (p.  135)  for  140  per  front 
foot.     After  paying  for  a  6-ft.   cement  sidewalk  costing -/ ^VS 
12)^  per  square  foot,  he  sold  the  lot  at  a  profit  of  $320. 
How  much  did  he  receive  for  it  ? 

5.  After  selling  60  acres  a  farmer  had  |  of  his  land  i  /  /;  ^ 
left.     Kow  many  acres  had  he  before  making  the  sale  ? 

6.  If  80  A.  of  land  cost  14000,  how  much  at  the  same^/  .,^ 
price  per  acre  will  320  A.  cost?  ,■ 

7.  If  hay  is  worth  $12  a  ton,  how  much  is  500  lb.  of^jL 
hay  worth  ?     How  much  is  400  lb.  worth  ?  '^;  • ' 

8.  If  a  boat  traveled  120  mi.  during  the  first  8  hr. 
after  leaving  port,  how  far  at  the  same  rate  will  it  travel 
in  1  da.?  in  2  da.?     ^^-^       ^^^ 

9.  At  $1  per  yard,  what  is  the  cost  of  J  yd.  of  silk?, 
off  yd.?  ofiyd.?,  7 


REVIEW  166 

10.  If  a  man's  expenses  for  3  mo.  amount  to  8135,  at 
the  same  rate,  how  much  will  his  expenses  amount  to  in 
lyr.?     ?5'^'C> 

11.  If  a  horse  is  fed  1  bu.  of  oats  in  5  da.,  how  many- 
bushels  will  be  necessary  to  feed  it  for  1  mo.  (30  da.)?  (> 

12.  If  it  costs  $10  to  pasture  6  horses  for  1  month,  how  ;  /)    < 
much  will  it  cost  to  pasture  9  horses  for  the  same  length    \ 
of  time  ? 

13.  At  1 7.25  per  ton,  how  much  will  6. 75  T.  of  coal  cost  ?  Vy .  ^^ 
0       14.    How  much  more  sugar  is  received  for  $  1  by  buying 

at  16 J  lb.  for  a  dollar  rather  than  at  6|^  per  pound  ?         T  ^ 

15.  On  the  morning  of  March  7  a  ship  captain  announced 
that  he  had  on  board  enough  provisions  to  last  80  da. 
Give  the  date  on  which  the  provisions  would  give  out. 

16.  A  boat  that  was  due  in  port  on  Dec.  25  arrived  on 
Jan.  6.     How  many  days  was  she  overdue  ?  ' 

17.  Change  to  decimals  ;    |,  J,  -J,  |,  -J,  ^,  |,  -J,  |,  -^,     /^ 
.,      li,  H,  li,  2f,  If. 

/^        18.    Change  to  common  fractions  :  .125,  .375,  .25,  .875, 
.6,  .625,  .8,  .40. 

19.  Find  .05  of  $200;   .06  of  $18.75;  1.04of|80.         ^ 

20.  $ 282  =.94  of ^-^;    $375  =.75  of ;    $60  =| 

of — .       '^^;^.ixv  0 


21.  160  =  1^  times  ^-^;   $60  =  1^   times 
=  1.20  of . 

22.  Change  to  lOOths  :    i,  f ,  i,  |,  |,  iV 

23.  If  4  of  the  cost  of  a  city  lot  is  $  1200,  how  much  is 
the  cost  of  the  lot  ?    '  -^  ^ 

24.  How  many  tons  of  hay  at  $12  a  ton  will  equal  in 
value  the  cost  of  laying  a  concrete  sidewalk  40  ft.  long  and 
6  ft.  wide  at  12^^  per  square  foot?   z.  /      ' 


X 


0  PART  III 


7- 


PERCENTAGE 

217.  Hundredths  as  Per  Cents. 

1.  Read  each  :   ^%  .60,  .10,  ^,  .06,  .85,  \^,  .05. 

2.  ^1^,  or  .05,  may  be  written  b^fo-  It  is  then  read  5 
per  cent.  Per  cent  means  hundredths.  5  per  cent  means 
5  of  the  100  equal  parts.  The  sign  (%)  is  called  the  per 
cent  sign. 

3.  The  unit  1  is  equivalent  to  how  many  hundredths  ? 
to  how  many  per  cent  ? 

4.  Read  the  following:  4%,  8%,  25%,  40%,  75%» 
100%,  150%,  200%,  6J%. 

5.  Express  as  per  cents  :  y^,  y^^,  ^-f^,  ^,  -^^  \^^. 

6.  Express  as  per  cents  :  .01,  .03,  .12,  .18,  .50,  .90, 
.99,  1,  2,  .125  (121%),  .375,  .625,  .875. 

7.  Write  as  common  fractions  :  7%,  2%,  40%,  85%, 
45%,  4%,  100%. 

8.  Write  as  decimal  fractions :  1%,  5%,  7%,  30%, 
3%,  75%,  80%,  T%,  100%,  37J%,  33^  %,  14f  %. 

218.  Finding  some  per  cent  of  a  number. 

1.  4%  of  $500  is  the  same  as  1500  multiplied  by  .04. 
Find  4%  of  $500;  of  $250;  of  $45.50;  of  $875. 

To  find  any  per  cent  of  a  number^  multiply  the  number  by 
the  required  per  cent  expressed  as  a  decimal  fraction. 

2.  Find  5%  of  $860;  of  $60;  of  $100;  of  $840.25. 

106 


HUNDREDTHS  AS  PER  CENTS  167 

3.  Find  12%  of  1400;  of  1350;  ofUlOO;  of  $247.25; 
of  11300. 

4.  Find  45%  of  650  mi. ;  80%  of  640  A. ;  62%  of  400 
bu.;  1%  of  $400. 

5.  Find  100%  of  $500.  Compare  100%  of  $500  with 
$500. 

6.  125%  means  -ifl,  or  1.25.     Find  125%  of  300  mi. 

7.  Name  a  per  cent  of  $600  that  is  the  same  as  $600; 
that  is  less  than  $600 ;  that  is  more  than  $600. 

8.  Is  80%  of  a  number  more  or  less  than  the  number? 
What  per  cent  of  a  number  is  equivalent  to  one  half  of 
the  number? 

9.  A  man  owes  8%  of  $700.  How  much  does  he 
owe? 

10.  A  man  borrowed  $800  and  agreed  to  pay  8%  of  the 
amount  borrowed  for  the  use  of  it  for  one  year.  How 
much  did  he  pay  for  the  use  of  $800  for  a  year? 

11.  A  man  borrowed  $700  and  agreed  to  pay  8%  of 
tlie  amount  borrowed  for  the  use  of  the  money  each  year. 
How  much  did  he  pay  for  the  use  of  $700  for  1  year? 
for  2  years?  for  3  years? 

12.  Money  paid  for  the  use  of  money  is  called  interest. 

13.  A  man  borrowed  $400  and  agreed  to  pay  6%  in- 
terest each  year.  How  much  interest  did  he  pay  in  1 
year?  in  ^  year  ?  in  1^  years  ?  in  2  years?  in  2^  years  ? 

14.  Find  the  interest  on  $600  for  2  years  at  6%. 

15.  A  real  estate  agent  sold  a  city  lot  for  Mr.  Thomas 
for  $1500.  He  received  for  his  services  5%  of  the  selling 
price  of  the  lot.  How  much  did  he  receive  for  selling  the 
lot? 


D 


168  PERCENTAGE 

16.  A  real  estate  agent  sold  a  city  lot  for  Mr.  Brown 
for  12000.  He  received  a  commission  of  6%  of  the  selling 
price  for  his  services.  How  much  did  he  receive  for  sell- 
ing the  lot?  How  much  did  Mr.  Brown  receive  for  the 
lot,  after  paying  the  commission? 

17.  A  farmer  shipped  25  tons  of  hay  to  a  commission 
merchant  in  a  city,  who  sold  it  for  $8  per  ton.  The  com- 
mission merchant  received  for  his  services  2%  of  the 
amount  of  the  sale.     Find  the  amount  of  his  commission. 

18.  A  commission  merchant  received  a  car  of  broom- 
corn  containing  8  tons,  which  he  sold  at  $120  per  ton. 
He  received  a  commission  of  5%  for  selling  it.  Find  the 
amount  of  his  commission. 

19.  A  farmer  shipped  40  tons  of  hay  to  a  commission 
merchant  who  sold  it  for  f  10  per  ton.  He  received  a 
commission  of  6%.  Find  the  amount  of  his  commission. 
How  much  did  the  farmer  receive  for  the  hay,  after  de- 
ducting the  commission? 

20.  A  farmer  had  160  acres  of  land.  He  sold  40%  of 
it.  How  many  acres  did  he  sell?  What  per  cent  of  the 
land  did  he  have  left?  If  he  received  1 85  per  acre  for 
the  land  sold,  how  much  did  he  receive  for  it? 

21.  A  farmer  had  320  acres  of  land.  He  sold  60%  of 
it  for  $80  per  acre  and  the  remainder  for  $75  per  acre. 
How  much  did  he  receive  for  the  land? 

22.  Mr.  Evans  borrowed  $250  of  Mr.  White  and  paid 
7%  interest.  At  the  end  of  the  year,  how  much  should 
Mr.  White  receive  from  Mr.  Evans,  if  he  received  the 
interest  and  the  money  loaned  ? 

23.  Find  50%  of  the  number  of  children  in  your  school- 
room. 


FRACTIONS  AS  PER  CENTS  169 

219.  Fractions  as  Per  Cents. 

1.  The  unit  1  is  equivalent  to  how  many  hundredths  ? 
to  how  many  per  cent  ? 

2.  What  per  cent  of  a  number  is  equivalent  to  the 
number  ?  to  J  of  the  number  ?  to  |  of  the  number  ?  to 
^^  of  the  number  ?  to  2  times  the  number  ?  to  5  times 
the  number  ? 

3.  State  how  a  common  fraction  may  be  reduced  to  a 
decimal  fraction. 

4.  Change  f  to  per  cent. 

go  __  gQ  of        Change  the  fraction  to  a  decimal, 
Model •         t^q  aq  extending  the  reduction  to  two  deci- 

mal places.     Express  hundredths  as 
per  cent. 
Another  Method:     Since  1  is  100%;    f  is   f   of  100%,  or  60%. 
20%      „ 
Work:  W%x|  =  60%. 

5.  Change  |  to  a  decimal  fraction.  .375  is  the  same 
as  .37 J,  which  is  the  same  as  Zl\  %. 

6.  State  how  a  common  fraction  may  be  changed  to 
hundredths,  expressed  as  a  decimal.  What  is  the  "mean- 
ing of  per  cent  ? 

To  change  a  common  fraction  to  per  cent^  divide  the  numera- 
tor hy  the  denominator^  and  carry  the  reduction  to  two  deci- 
mal places  in  the  quotient.    Express  the  quotient  as  per  cent, 

7.  Change  to  per  cents :  |,  |,  |,  |-,  |,  ^9_  1, 

8.  Change  to  per  cents :  11,  if,  -j^^  |f  if,  |f. 

9.  Change  to  per  cent :  1|. 

1.80  =  180  (fo 
Model:      1-|  =  |.     5)9.00 

la  Change  to  per  cents :  IJ,  If,  1\,  IJ,  If,  1|. 


ro 

PERCENTAGE 

11. 

Memorize  the  following : 

1  = 

100% 

1  =  60  % 

T%=    30% 

i  =  12J% 

1  = 

50% 

1=80% 

T^=    70% 

f  =  S7^% 

i  = 

25% 

2V=    5% 

A=    90% 

f=62J% 

1  = 

75% 

2V=    4% 

iV=    8i% 

i  =  87J  % 

i  = 

20% 

3V=    2% 

i  =  S3i  % 

i  =  16|% 

1  = 

40% 

tV  =  io% 

|  =  66|% 

|  =  14f% 

220.   Oral  Exercises. 

1.  Findl6|  %  of  $480. 

16f  %  of  $480  may  be  found  by  multiplying  $480  by  .16f,  or  it 
may  be  found  by  taking  ^  of  $  480.  Solve  by  both  methods.  Which 
method  is  the  shorter  ? 

2.  Certain  per  cents  of  quantities  may  be  found  more 
easily  by  the  use  of  fractional  equivalents.  One  of  these 
is  33|  %.     Name  others. 

Solve,  using  fractional  equivalents :  * 

3.  16|  %  of  24  hr.  12.  62i  %  of  640  A. 

4.  331  %  of  1 15.  13.  87|-  %  of  320  rd. 

5.  142  %  of  1 35.  14.  16f  %  of  $  7.20. 

6.  66f  %  of  36  mi.  15.  14f  %  of  184. 

7.  37|  %  of  48  yd.  16.  66|  %  of  60  bu. 

8.  25  %  of  320  rd.  17.  33 J  %  of  12.10. 

9.  50  %  of  11.60.  18.  75  %  cf  1400. 

10.  8J  %  of  360  da.  19.   80  %  of  1 25. 

11.  12|  %  of  80)2^.  20.   16|  %  of  30  ft. 

*  This  exercise  should  be  supplemented  with  oral  drills  until  the  pupils 
are  able  to  find  the  above  per  cents  readily  by  the  use  of  their  fractional 
equivalents.  The  fractional  equivalents  of  the  above  per  cents  should  be 
used  in  subsequent  exercises. 


S. 


DISCOUNT  171 

221.  Oral  Exercises. 

1.  At  a  sale  the  following  discounts  were  advertised. 
(a)  Find  the  amount  of  reduction  and  (5)  the  selling 
price : 

a.  16f  %  off  on  carpets  marked  90^  per  yard. 

h.  33^  %  off  on  bric-a-brac  marked  f  6. 

c,  14f  %  off  on  ladies'  hats  marked  $14. 

d,  66|  %  off  on  damaged  cloth  marked  30^  per  yard. 

e,  Ill  %  off  on  tables  marked  $16. 
/.  371  %  off  on  cloaks  marked  $16. 

2.  At  what  price  should  the  following  be  marked  : 

a.  Cloth  that  cost  80^  per  yard,  to  make  a  profit  of 
2.5%? 

h.    Suits  that  cost  $15,  to  make  a  profit  of  33^  %  ? 

c.  Hats  that  cost  $2.40,  to  make  a  profit  of  25%  ? 

d.  Shoes  that  cost  $3  per  pair,  to  make  a  profit  of 
331%? 

g.    Silk  that  cost  $1.50,  to  make  a  profit  of  50  %  ? 

/.    Overcoats  that  cost  $16,  to  make  a  profit  of  37^%? 

g.    Lace  at  60  ^  per  yard,  to  make  a  profit  of  16|  %  ? 

3.  What  per  cent  of  a  number  remains  after  subtract- 
ing 25%  of  it?  20%?  40%?  75%?  5%?  66|%?  33i%? 
50  %  ?  60  %  ?  10  %  ?  2  %  ?  100  %  ?  90  %  ? 

4.  What  fractional  part  of  a  quantity  remains  after 
subtracting  50%  of  it?  20%?  25%?  30%?  40%?  75%? 
33|-%?80%?66f%?10%?  16f%?  142%?  5^^?  121%? 
100%?  90%?  37J%?  15%? 

5.  How  much  remains  of  $24  after  deducting  50%  of 

it?  25%?  75%?  16|%?  331%?  66|-%?  12i%?  37|%? 

6.  How  much  remains  of  $36  after  deducting  25%  of 
it?  50%?  75%?  33J%?  66f%?  16|%?  100%? 


172  PERCENTAGE 

7.  Find  1  of  36;  50%  of  80;  ^  of  90;  83J%  of  76; 
^  of  200;  75%  of  200;  20%  of  15  •  |-  of  60;  40%  of 
120;  ^  of  40  ;  12J%  of  72  ;  66^%  of  90  ;  f  of  64. 

8.  A  merchant  bought  silk  at  $1.80  per  yard  and  sold 
it  at  a  profit  of  33J  % .  How  much  did  he  make  on  each 
yard  ? 

9.  A  man  bought  hay  at  1 8  per  ton  and  sold  it  at  a 
profit  of  25  % .  What  was  his  profit  on  each  ton  ?  "What 
was  the  selling  price  per  ton  ? 

10.  A  grocer  bought  tea  at  40^  per  pound  and  sold  it 
at  a  profit  of  50  %.     What  was  the  selling  price? 

11.  A  suit  of  clothes  marked  $  20  was  sold  at  a  reduc- 
tion of  20  %.  Find  the  amount  of  the  discount  and  the 
selling  price  of  the  suit. 

12.  A  wagon  that  cost  $72  was  sold  at  a  profit  of 
16|  %.     What  was  the  selling  price  of  the  wagon  ? 

""^^^  13-  A  merchant  advertised  a  reduction  of  25  %  on  all 
goods.  Find  the  reduction  on  suits  marked  $  30  ;  on 
shoes  marked  $4  ;  on  hats  marked  $2;  on  cloth  marked 
80  ^  per  yard  ;  on  rugs  marked  $  6. 

14.  A  house  owned  by  Mr.  West  was  rented  to  Mr. 
James  by  a  real  estate  firm  for  one  year  at  $  30  per  month. 
If  the  firm  received  as  commission  10  %  of  the  first 
month's  rent,  what  was  the  amount  of  the  commission  ? 

15.  A  hardware  merchant  invested  $  5000  in  his  busi- 
ness. He  cleared  15  %  on  the  investment  in  one  year. 
What  was  the  amount  cleared  during  the  year  ? 

16.  In  the  catalogue  of  a  carriage  manufacturer  a  cer- 
tain carriage  was  listed  at  $  150.  It  was  bought  by  a  re- 
tail dealer  at  a  discount  of  20  %  from  the  list  price.  How 
much  did  the  carriage  cost  the  retail  dealer  ? 


PERCENTAGE  178 

17.  A  farmer  shipped  50  boxes  of  apples  to  a  commis- 
sion merchant,  who  sold  them  at  90  i  per  box.  The  com- 
mission merchant  charged  a  commission  of  5  %  for  his 
services.  Find  the  amount  of  his  commission.  He  paid 
freight  charges  amounting  to  $3.50.  How  much  should 
he  remit  to  the  farmer  after  deducting  for  commission 
and  freight  ? 

18.  Mr.  A  bought  a  cow  for  140  and  sold  it  at  a 
profit  of  20  % .     What  was  the  selling  price  of  the  cow  ? 

19.  Mr.  A  sold  a  cow  for  |-  of  the  cost.  He  received 
%  48  for  the  cow.     Find  the  cost  of  the  cow. 

20.  Mr.  A  sold  a  cow  at  a  profit  of  \  of  the  cost.  His 
profit  was  %  8.     Find  the  cost. 

21.  A  real  estate  dealer  bought  a  lot  for  %  600.  After 
five  years  he  sold  it  at  a  profit  of  100  %  of  the  cost.  What 
per  cent  of  the  cost  of  the  lot  did  he  receive  for  it  ? 

(22  J  A  merchant's  stock  of  goods  valued  at  %  4500  was 
damaged  by  fire.  He  was  obliged  to  dispose  of  the  goods 
for  ^^\  %  of  their  former  value.  What  fractional  part  of 
their  value  did  he  receive  for  them  ?  How  much  did  he 
receive  for  his  stock  ? 

23.  A  dealer  was  asked  the  price  of  a  certain  carriage. 
He  replied  that  he  would  sell  the  carriage  for  %  200  and 
allow  the  purchaser  60  days  in  which  to  make  the  pay- 
ment, or  that  he  would  allow  a  discount  of  2  %  for  cash 
payment.     Find  the  cash  price  of  the  carriage. 

24.  Mr.  James  pays  \\%  taxes  on  $4000.  Find  the 
amount  of  his  tax. 

25.  A  man  bought  a  lot  for  f  1600.  He  sold  it  for 
%  1800.  How  much  did  he  gain  on  the  lot  ?  His  gain 
was  what  part  of  the  cost  \ 


174  PERCENTAGE 


■f" 


26.  Frank  Thomas  borrowed  $  1200  for  1  yr.  at  6% 
interest.  How  much  did  he  pay  for  the  use  of  the 
money  ? 


222.  Finding  the  number  of  which  a  given  number  is  a 
certain  per  cent. 

1.  If  4  %  of  a  sum  of  money  is  $12,  what  is  1  %  of  it? 
If  1  %  of  a  sum  of  money  is  $3,  what  is  100  %  of  it? 

2.  When  5%  of  a  selling  price  is  f  80,  what  is  1  %  of 
the  selling  price  ?     What  is  the  selling  price  ? 

3.  If  8  %  of  a  number  is  160,  what  is  1  %  of  the  num- 
ber?   What  is  2  %  of  the  number?    What  is  the  number? 

4.  When  6  %  of  a  number  is  given,  how  may  1  %  of  it 
be  found?  How,  then,  may  the  number  be  found?  Any 
number  is  equivalent  to  how  many  per  cent  of  itself? 

5.  To  find  a  number  when  a  certain  per  cent  of  it  is 
given,  find  what  1  %  of  it  is,  then  find  100  %  of  it.  By 
this  method  find  the  number  of  which  24  is  8  % . 

6.  When  the  multiplier  and  the  product  are  given, 
how  may  the  multiplicand  be  found? 

7.  Some  number  when  multiplied  by  4  is  216,  what  is 
the  number?  Some  number  when  multiplied  by  .08  is 
24,  what  is  the  number? 

8.  To  say  that  7  %  of  a  number  is  161,  is  the  same  as 
to  say  that  some  number  when  multiplied  by  .07  gives 
161  as  a  product.  The  number  may  be  found  by  divid- 
ing 161  by  .07. 

To  find  the  number  of  which  a  given  number  is  a  certain 
per  cent,  divide  the  given  number  by  the  given  per  cerU 
expressed  as  a  decimal. 


PERCENTAGE  175 

In  each  of  the  following,  name  the  multiplier  and  the 
product,  and  state  how  the  multiplicand  may  be  found : 

9.  15  %  of  a  number  is   60.    12.  8  %  of  some  land  is  25.6  A. 

10.  9  %  of  a  number  is  135.    13.  6  %  of  some  money  is  $  108. 

11.  18  %  of  a  number  is   81.    14.  45  %  of  a  crop  is  135  bu. 

223.  Written  Exercises. 

1.  $11.49  is  12%  of  what  amount? 

Model  •  S345  75         Some  amount  when  multiplied  by 

'. —     .12  gives  $41.49  as  a  product.     The 

.l^^-lt>4i.4y  amount  is  found  by  dividing  the 

product  (141.49)  by  the  multiplier  (.12). 

2.  240  mi.  is  12  %  of  how  many  miles  ? 

3.  128  tons  is  8  %  of  how  many  tons? 

4.  A  man  paid  $32  interest  for  the  use  of  some  money 
for  one  year  at  8  % .     What  was  the  sum  borrowed  ? 

5.  A  farmer  received  40%  of  a  crop  as  rent  for  his 
land.  His  share  of  the  wheat  amounted  to  400  bu.  in 
one  year.  What  amount  of  wheat  was  raised  on  the  farm 
in  that  year? 

6.  A  man  received  |80  interest  on  some  money  which 
he  loaned  for  a  year  at  10%.  Find  the  amount  of  the 
loan. 

7.  On  a  certain  day  4  of  the  pupils  in  a  school  were 
absent.  This  was  8%  of  the  number  enrolled.  How 
many  pupils  were  enrolled  in  the  school? 

8.  During  one  season  a  baseball  team  lost  14  games, 
which  was  40%  of  the  number  of  games  played.  How 
many  games  did  the  team  play? 


O 


X 


176  PERCENTAGE 

9.  Two  men  entered  into  partnership  in  a  retail  hard- 
ware store.  One  agreed  to  furnish  40%  of  the  capital 
and  the  other  60  %  of  the  capital.  The  partner  who  con- 
tributed 40%  of  the  capital  invested  $2400.  Find  the 
whole  amount  of  the  capital. 

10.  A  farmer  sold  120  acres  of  land,  which  was  30% 
of  his  entire  farm.  How  many  acres  had  he  before  mak- 
ing the  sale?    What  per  cent  of  his  farm  did  he  still  own? 

224.  Oral  Exercises. 

1.  A  number  is  how  many  times  20%  of  itself?  If 
20  %  of  a  number  is  8,  what  is  the  number  ? 

2.  A  number  is  how  many  times  33  J  %  of  itself?  If 
33 J  %  of  a  number  is  25,  what  is  the  number? 

3.  What  part  of  a  number  is  each  of  the  following 
per  cents  of  the  number :  25  %,  37^  %,  50  %,  62|  %,  66|%, 
75%,  871%,  16|%,  14f  %,  8J%,  12|%,  20%,  33J%? 

4.  A  number  is  how  many  times  each  of  the  following 
per  cents  of  itself:  14f%,  75%,  25%,  16|%,  50%,  81%, 
20%,  371%,  12|%,  331%,  62J%,  871%,  66|%? 

5.  If  50  %  of  the  amount  of  money  a  boy  has  is  f  12, 
how  much  money  has  he?  How  much  money  has  he  if 
25%  of  his  money  is  |5?  if  10%  of  his  money  is  f  3  ? 
if  33^%  of  his  money  is  $8?  if  16^  %  of  his  money  is 
1 10  ?  if  66|  %  of  his  money  is  $  24  ? 

6.  $  5  is  25  %  of  —.  $8  is  33^  %  of  — .  4  mi.  is  20  % 
of  — .  6  gal.  is  50  %  of  — .  12  yd.  is  75%  of  — .  160 
rd.  is  10  %  of  — . 

7.  Find  the  number  of  which  16  is  25  %  ;  30  is  20  %  ; 
18  is  66^  %  ;  40  is  200  %  ;  60  is  300  %  ;  15  is  37J  %  ; 
25  is  50  %  ;  50  is  62^  %  ;  70  is  33J  %  ;  75  is  100  %. 


11 

r 


PERCENTAGE  177 

8.  When  14|  %  of  a  number  is  given,  how  may  the 
number  be  found?  14 1  %  of  a  farm  is  25  acres.  How 
many  acres  are  there  in  the  farm  ? 

9.  A  man  sold  45  head  of  cattle,  which  was  25%  of 
the  number  he  had.     How  many  head  of  cattle  had  he? 

10.  A  merchant  sold  goods  at  a  discount  of  16|  %  from 
the  cost  price  and  lost  $60.     What  was  the  cost? 

11.  In  37 1  %  of  a  farm  there  are  90  acres.  How  many 
acres  are  there  in  the  farm? 

12.  A  merchant  made  12|  %  on  the  cost  of  some  goods 
by  selling  them  at  a  profit  of  $6.  Find  the  cost  of  the 
goods.     Find  tlie  selling  price  of  the  goods. 

13.  A  number  is  how  many  times  ^  of  itself?  |-  of 
itself?  I  of  itself?  f  of  itself? 

14.  If  66|%  of  a  number  is  120,  what  is  the  number? 

15.  If  87 J  %  of  a  number  is  70,  what  is  the  number? 

16.  In  a  spelling  test  a  boy  spelled  correctly  30  words, 
which  was  75  %  of  the  number  of  words  in  the  test.  Find 
the  number  of  words  in  the  test. 

17.  Mr.  A  sold  a  cow  at  a  profit  of  25%.  His  profit 
amounted  to  f  10.     Find  the  cost. 

18.  Mr.  A  sold  a  cow  at  a  loss  of  25  %  of  the  cost. 
For  what  part  of  the  cost  did  he  sell  the  cow?  He  re- 
ceived $  30  for  the  cow.     Find  the  cost. 

19.  A  fruit  grower  planted  120  apple  trees.  20  of 
them  died.     What  per  cent  of  the  trees  died  ? 

20.  If  12  trees  are  25  %  of  the  number  planted  by  a 
fruit  grower,  how  many  trees  did  he  plant  ? 

21.  Eight  pupils  were  absent  from  school  on  a  certain 
day,  which  was  20  %  of  the  pupils  enrolled.  How  many 
pupils  were  enrolled  in  the  school  ? 

AK.  —  12 


D 


178  PERCENTAGE 

225.   Written  Exercises. 

1.  A  f iirin  was  sold  for  f  6000,  which  was  25  %  more 
than  it  cost.     Find  the  cost  of  the  farm. 

The  fractional  equivalents  of  per  cents  should  be  used  whenever 
the  work  can  be  made  easier  or  shorter  by  their  use. 

Model  A  :  f  of  the  cost  of  the  farm  =  $  6000. 

$1200 
I  of  the  cost  of  the  farm  =  i  of  ^jim,  or  $  4800. 

Since  the  farm  was  sold  for  f  (125%)  of  its  cost,  the  cost  of  the 
farm  is  |  of  the  seUing  price. 

Since  $6000  is  125%  of  the  cost  of 
f  48  00.      the  farm,  the  cost  of  the  farm  may  be 
Model  B  :  1.25)^6000.00      found  by  dividing  1 6000  by  1.25.    (See 
Sec.  223.) 

2.  A  city  lot  was  sold  for  $1200,  which  was  20  %  more 
than  it  cost.     Find  the  cost  of  the  lot. 

3.  After  increasing  his  stock  33J%,  a  merchant  found 
that  he  had  $12,000  invested.  Find  the  amount  of  his 
investment  before  the  increase. 

4.  A  sum  of  money  was  borrowed  for  a  year  at  8  % 
interest.  At  the  end  of  the  year  the  money  borrowed  and 
the  interest  amounted  to  $432.  What  per  cent  was  this 
of  the  sum  borrowed  ?     Find  the  sum  borrowed. 

5.  If  the  population  of  a  certain  city  in  1905  was  81,250, 
and  this  was  an  increase  of  25  %  over  the  population  in 
1895,  what  was  the  population  in  1895? 

6.  A  dealer  sold  a  carriage  for  $96,  at  a  loss  of  20  %. 
What  per  cent  of  the  cost  of  the  carriage  did  he  receive 
for  it  ?     How  much  did  the  carriage  cost  him  ? 

7.  A  firm  sold  a  carriage  to  a  retail  dealer  for  $119, 
which  was  at  a  discount  of  15%  from  the  list  price  of 
the  carriage.     Find  the  list  price  of  the  carriage. 


-^•^ 


PERCENTAGE  179 

226.   Finding  what  per  cent  one  number  is  of  another. 

1.  Each  of  the  following  fractions  is  equivalent  to  what 
per  cent ;  J,  -J,  |,  |,  |,  f ,  -|,  -J,  |,  J,  f  J,  f ,  |,  |,  ^j,  -^q,  -^, 

r5^  A'  2^'  A  ^ 

2.  Each  of  the  following  is  equivalent  to  what  per  cent : 

ij,  li,  i{,  H,  If,  If,  H,  li,  H,  i^v  If.  If'  If  If  i|  ? 

3.  8  is  what  part  of  16  ?  What  is  the  ratio  of  8  to  16  ? 
1^  of  a  number  is  what  per  cent  of  the  number  ?  8  is  what 
per  cent  of  16  ? 

4.  24  is  what  part  of  36  ?    24  is  what  per  cent  of  36  ? 

5.  12  is  what  per  cent  of  36  ?  of  24  ?  of  48  ?  of  60  ? 

6.  Express  as  a  common  fraction  the  ratio  of  6  to  8 ; 
of  20  to  25 ;  of  25  to  20  ;  of  30  to  35 ;  of  40  to  60. 

7.  Express  as  hundredths  in  decimal  form  the  ratio  of 
3  to  5  ;  of  5  to  8 ;  of  4  to  5  ;  of  24  to  30. 

8.  16  is  what  per  cent  of  20  ? 

^6  =  |.    Reduce  f  to  a  decimal  and 

.80  =  80  %       carry  the   reduction   to  two  decimal 

Model  A :  5)4.00  places  in  the  quotient.    .80  is  the  same 

as  80%. 

Some  per  cent  of  20  is  16.     20  is 

the  multiplicand  and  16  is  the  prod- 

^-  ^"  '°  uct.      The  multiplier  may  be  found 

Model  B  :  20)16.00  ^^  ^.^.^^^  ^^^  p^^^^^^  ^^g^  ^^  ^^^ 

multiplicand   (20).      The   multiplier 
is  .80,  which  is  the  same  as  80  %. 

To  find  what  per  cent  one  number  is  of  another,  express  as 
a  common  fraction  the  ratio  of  the  one  to  the  other,  and  reduce 
the  fraction  to  a  decimal,  carrying  the  reduction  to  two 
decimal  places  in  the  quotient.  Express  the  result  as  per 
cent. 


3. 

50  is  20. 

4. 

25  is  50. 

5. 

18  is  15. 

6. 

48  is  60. 

7. 

54  is  27. 

8. 

240  mi.  is  180  mi. 

9. 

360  bu.  is  600  bu. 

180  PERCENTAGE 

227.  Written  Exercises. 

1.  What  per  cent  of  120  mi.  is  90  mi.  ? 

2.  145  is  what  per  cent  of  $50  ? 

Find  what  per  cent  of : 

10.  320  rd.  is  80  rd. 

11.  640  A.  is  120  A. 

12.  360  da.  is  30  da. 

13.  5280  ft.  is  1760  ft. 

14.  5000  ft.  is  1000  ft. 

15.  2000  mi.  is  6000  mi. 

16.  2000  lb.  is  750  lb. 

17.  A  man  owned  320  A.  of  land.  He  sold  80  A. 
What  per  cent  of  his  land  did  he  sell  ?  What  per  cent  of 
it  did  he  have  left? 

18.  A  coal  dealer  bought  240  tons  of  coal.  He  sold 
160  tons.  What  per  cent  of  it  did  he  sell?  What  per 
cent  of  it  did  he  have  left  ? 

228.  Oral  Exercises. 

1.  1^  times  a  number  is  what  per  cent  of  the  number  ? 
If  1^  times  a  number  is  36,  what  is  the  number?  If 
133 J  9^  of  a  number  is  48,  what  is  the  number  ? 

2.  What  per  cent  of  a  number  is  1 J  times  the  number  ? 
If  150  %  of  a  number  is  12,  what  is  the  number  ? 

3.  If  If  times  a  number  is  20,  what  is  the  number  ?  If 
166^%  of  a  number  is  60,  what  is  the  number? 

4.  If  6  %  of  a  certain  amount  is  $30,  what  is  1  %  of  the 
amount  ? 


GAIN  AND  LOSS  181 

229.  Per  Cent  of  Gain  or  Loss. 

1.  Mr.  A  bought  a  cow  for  $40  and  sold  it  at  a  gain  of 
$S,     $8,  the  gain,  is  what  per  cent  of  f  40,  the  cost  ? 

2.  Mr.  Clark  bought  a  cow  for  $40  and  sold  it  for  $48. 
Find  the  gain.     The  gain  is  what  per  cent  of  the  cost  ? 

3.  Mr.  Brown  bought  a  horse  for  $120  and  sold  it  for 
$100.  Find  the  amount  of  his  loss.  His  loss  is  what  per 
cent  of  the  cost  of  the  horse  ? 

4.  When  the  cost  price  and  the  selling  price  are  given, 
how  is  the  amount  of  the  gain  or  loss  found  ? 

5.  The  per  cent  which  the  amount  of  gain  or  loss  is  of 
the  cost  is  called  the  gain  or  loss  per  cent.  The  gain  or 
loss  per  cent  is  always  some  per  cent  of  the  cost. 

To  find  the  gain  or  loss  per  cent,  find  what  per  cent  the 
amount  of  gain  or  loss  is  of  the  cost. 

6.  A  furniture  dealer  bought  some  rocking-chairs  for 
$4  each  and  sold  them  for  $6  each.  How  much  did  he 
make  on  each  chair  ?     What  was  his  gain  per  cent  ? 

7.  A  bicycle  that  cost  $40  was  sold  for  $30.  What 
was  the  loss  per  cent  ? 

8.  A  fruit  dealer  bought  berries  at  6^  per  box  and 
sold  them  at  10  ^  per  box.     What  was  his  gain  per  cent  ? 

9.  A  man  bought  a  cow  for  $30  and  sold  it  for  $40. 
What  was  his  gain  per  cent  ? 

10.  A  newsboy  bought  papers  for  Sfi  each  and  sold  them 
for  5  ^  each.     What  was  his  gain  per  cent  ? 

11.  A   newsboy  bought  papers   for   1^  each  and  sold 
them  for  2  ^  each.     What  was  his  gain  per  cent  ? 


182 


PERCENTAGE 


230.   Oral  Exercises. 

Find  the  gain  or  loss  per  cent 


Cost 

Selling  Price 

Cost 

Gain 

Loss 

1. 

$10 

$15 

6.    $16 

$4 

2. 

$15 

$10 

7.    $12 

•4 

3. 

$25 

$30 

8.    $15 

$3 

4. 

$30 

$25 

9.    $20 

$2 

5. 

$40 

$45 

10.   $25 

«5 

231.   Written  Exercises. 

Find  the  value  of  x  in  each 


Cost 

Selling  Price 

Gain 

Loss 

Gain% 

Loss% 

1.   $80 

$100 

X 

X 

2.   $75 

X 

$25 

X 

3.       X 

$120 

$30 

X 

4.   $50 

X 

$5 

X 

5.      X 

$60 

$20 

X 

6.      X 

$4.80 

$1.20 

X 

7.   $20 

X 

X 

8% 

8.   $36 

X 

X 

hfo 

9.      X 

$80 

X 

20% 

10.      X 

■      $24 

X 

20% 

11.  Ex.  1  above  may  be  stated  in  the  form  of  a  prob- 
lem, thus  :  A  man  bought  a  horse  for  $  80  and  sold  it  for 
$100.  Find  the  gain  or  loss  per  cent.  State  problems 
for  Exs.  1-10  above. 

12.  A  certain  baseball  team  won  6  games  out  of  10. 
What  per  cent  of  the  games  did  the  team  win  ? 


GAIN  AND  LOSS  188 

232.  Written  Exercises. 

1.  A  real  estate  agent  bought  a  city  lot  for  11200  and 
sold  it  for  §1500.     What  was  the  gain  per  cent  ? 

2.  A  merchant  disposed  of  a  stock  of  goods  valued  at 
18000  for  $6000.     What  was  the  loss  per  cent? 

3.  An  agent  received  $40  for  selling  hay  at  a  com- 
mission of  5%.     Find  the  selling  price  of  the  hay. 

4.  The  interest  on  a  sum  of  money  for  one  year  at 
6  %  was  $  72.     On  what  amount  was  interest  paid  ? 

5.  A  farmer  lost  45  %  of  his  wheat  crop  by  fire.  His 
loss  amounted  to  600  bushels.  What  was  the  amount  of 
his  entire  crop? 

6.  After  suffering  a  loss  of  35%  of  the  value  of  his 
stock  of  goods,  a  merchant  found  that  the  remainder  of 
his  stock  was  worth  $13,000.  What  was  the  value  of  his 
stock  before  the  loss  ? 

7.  A  stock  of  goods  valued  at  $4500  was  partly  de- 
stroyed by  fire.  After  the  fire  the  stock  was  estimated 
to  be  worth  $3000.     What  was  the  per  cent  of  loss  ? 

8.  Mr.  Thomas  bought  a  farm  for  $5250.  He  rented 
the  farm  for  $420  a  year.  His  rent  amounted  to  what 
per  cent  of  his  investment  ? 

9.  Mr.  Bunker  bought  a  lot  for  $1500  and  built  a 
house  on  it  costing  $3000.  He  rented  his  property  for 
$300  a  year.  His  rent  amounted  to  what  per  cent  of  his 
investment  ? 

10.  A  business  block  in  a  city  was  advertised  for  sale 
for  $75,000.  This  block  rented  for  $500  per  month. 
The  income  from  the  rent  amounted  to  what  per  cent  of 
the  price  asked  for  the  property  ? 


> 


184  PERCENTAGE 

REVIEW 
233.  Oral  Exercises. 

1.  By  selling  land  at  $25  per  acre  more  than  it  cost 
him,  a  farmer  gained  20  %  of  the  cost  of  the  land.  Find 
the  cost  of  the  land. 

The  gain,  or  $  25  per  acre,  amounts  to  20  %  of  the  cost,  or  |  of  the 
cost.  Since  $  25  per  acre  is  ^  of  the  cost,  the  cost  is  5  times  ^25  per 
acre,  or  $  125  per  acre. 

2.  By  selling  a  carriage  for  $15  more  than  it  cost 
him,  a  dealer  gained  12J  %  of  the  cost  of  the  carriage. 
Find  the  cost  of  the  carriage. 

3.  A  city  lot  increased  $200  in  value,  which  amounted 
to  an  increase  of  33  J  %  of  its  cost.  Find  the  cost  of  the 
lot. 

4.  A  gain  of  66^%  of  the  cost  amounted  to  a  gain 
of  $120.     Find  the  cost. 

5.  A  horse  was  sold  for  $150,  which  was  120  %  (|)  of 
the  cost.     Find  the  cost  of  the  horse. 

6.  By  selling  an  overcoat  for  $35,  a  merchant  made  a 
profit  of  16|  %  of  the  cost.  What  fraction  expresses  the 
ratio  of  the  selling  price  to  the  cost  ?     Find  the  cost. 

7.  A  boy  sold  a  pony  for  $6  more  than  it  cost  him. 
His  profit  amounted  to  16|  %  of  the  cost  of  the  pony. 
Find  the  cost  and  the  selling  price. 

8.  After  selling  80%  of  his  land,  a  farmer  had  what 
per  cent  of  it  left  ?  After  selling  80  %  of  his  land,  a 
farmer  had  left  40  acres.  How  many  acres  had  he  before 
making  the  sale  ? 

9.  By  selling  a  cow  for  $32,  a  farmer  lost  20%  of  the 
cost  price.  What  fraction  expresses  the  ratio  of  the  sell- 
ing price  to  the  cost  ?     Find  the  cost  of  the  cow. 


Y- 


REYIEW  186 

10.  A  liveryman  made  40  %  on  the  cost  of  a  horse  by 
selling  the  horse  for  $140.  What  fraction  expresses  the 
ratio  of  the  selling  price  to  the  cost  ?     Find  the  cost. 

11.  By  selling  a  lot  for  1640,  a  dealer  lost  20%  of  the 
cost  price.  The  selling  price  was  what  fraction  of  the 
cost  of  the  lot  ?     Find  the  cost  of  the  lot. 

12.  A  field  of  wheat  was  damaged  by  floods  to  the  ex- 
tent of  25  %  of  the  expected  yield.  The  yield  amounted 
to  30  bushels  of  oats  to  the  acre.  This  was  what 
fractional  part  of  the  expected  yield?  What  was  the 
expected  yield  ? 

13.  A  watch  that  cost  $25  was  sold  for  200  %  of  the 
cost.     Find  the  selling  price  of  the  watch. 

14.  A  painting  that  cost  f  60  was  sold  for  33^  %  less 
than  it  cost.  It  was  sold  for  what  fractional  part  of  its 
cost  ?     Find  the  selling  price.  V" 

15.  A  merchant  made  a  profit  of  25  %  of  the  cost  of 
silk  by  selling  it  for  | .  80  per  yard.  Find  the  cost  of  the 
silk  per  yard. 

16.  A  sum  of  money  loaned  at  7  %  yields  f  42  interest 
each  year.     Find  the  sum  loaned. 

17.  $  20  is  what  part  of  %  100  ?  A  carriage  that  cost 
%  100  was  sold  for  $120.  It  was  sold  for  what  per  cent 
of  the  cost  price  ? 

18.  A  stove  that  costs  $40  is  sold  for  1 36.  The  loss 
is  what  part  of  the  cost  of  the  stove  ?  The  loss  is  what 
per  cent  of  the  cost  of  the  stove  ?  The  selling  price  is 
what  per  cent  of  the  cost  price  ? 

19.  A  farm  that  costs  %  60  per  acre  is  sold  for  $  70 
per  acre.  The  gain  on  each  acre  is  what  part  of  the  cost 
per  acre  ?     The  gain  is  what  per  cent  of  the  cost  ? 


"b 


18«  PERCENTAGE 

234.   Written  Exercises. 

1.  Hay  that  cost  $40  for  5  tons  was  sold  at  $  9  a  ton. 
What  was  the  profit  on  each  ton?  the  gain  per  cent? 

2.  A  clothing  merchant  advertised  a  reduction  of  20  % 
on  all  goods.  Find  the  amount  of  reduction  and  the  sale 
price  of  suits  marked  $35,  hats  marked  §2,  suspenders 
marked  50^,  shoes  marked  $3.50,  neckties  marked  25^, 
overcoats  marked  $20,  cuffs  marked  20^  per  pair,  collars 
marked  2  for  25^. 

3.  A  jeweler  sold  a  watch  for  $26,  which  was  at  a 
profit  of  33^%.     Find  the  cost  of  the  watch. 

4.  Goods  damaged  by  fire  were  sold  for  $2400,  which 
was  at  a  loss  of  40  % .     What  was  their  original  value  ? 

5.  What  per  cent  of  his  earnings  does  a  man  save  who 
earns  $  80  a  month  and  saves  $300  each  year  ? 

6.  A  farmer  paid  $4000  for  a  farm  and  sold  it  for 
$  4200.     Find  the  gain  per  cent. 

7.  A  man's  yearly  income  from  a  farm  valued  at  $  6000 
is  $  1500.  The  income  is  what  per  cent  of  the  value  of 
the  farm? 

8.  By  selling  a  carriage  for  12 J  %  more  than  it  cost 
him,  a  dealer  made  a  profit  of  $15.  How  much  did  the 
carriage  cost  him  ? 

9.  Two  men  entered  into  partnership  to  purchase  a 
boat  that  cost  $300.  Each  contributed  one  half  of  the 
capital.  One  of  the  men  sold  his  share  of  the  boat  for 
$120.     Did  he  gain  or  lose,  and  what  per  cent? 

10.  A  house  that  was  valued  at  $  2400  was  rented  so 
that  the  yearly  rent  amounted  to  12  %  of  the  value  of  the 
property.     What  was  the  monthly  rent  of  the  house  ? 


^ 


REVIEW  187 

11.  By  selling  a  cow  for  $15  more  than  it  cost  him,  a 
farmer  gained  33^  %  of  the  cost  of  the  cow.  Find  the  cost 
of  the  cow.     Find  the  selling  price. 

12.  Tea  that  was  sold  at  60  f^  per  pound  was  sold  at  a 
profit  of  33J%.     Find  the  cost  of  the  tea. 

13.  A  piano  dealer  sold  two  pianos  for  $240  each.  On 
one  he  made  a  profit  of  20  %  and  on  the  other  he  lost  20  %, 
How  much  did  each  of  the  pianos  cost  him  ?  Did  he  gain 
or  lose  on  the  two  pianos  ? 

14.  How  should  goods  that  cost  f  1.20  per  yard  be 
marked  to  sell  at  a  profit  of  20%  ?  25%  ?  33^%  ?  50%  ? 

15.  Three  men  bought  some  land  for  $  3600.  One  fur- 
nished $1500,  another  $1200,  and  the  third  1 900.  They 
sold  the  land  for  $4200.  What  per  cent  of  the  capital 
did  each  furnish?  What  per  cent  of  the  profit  should 
each  receive  ?     What  was  each  man's  share  of  the  profit  ? 

16.  A  man  had  $800  in  a  bank.  He  drew  out  first 
$200  and  then  $300.  What  per  cent  of  his  money  did  he 
draw  out  ?     What  per  cent  was  left  in  the  bank  ? 

17.  The  salary  of  a  clerk  was  increased  from  $60  per 
month  tp  $75  per  month.  What  per  cent  of  increase  was 
made  in  his  salary  ?  The  increase  would  amount  to  how 
many  dollars  in  two  years  ? 

18.  The  population  of  Los  Angeles  was  50,300  in  1890 
and  102,479  in  1900.  The  population  in  1900  was  what 
per  cent  of  the  population  in  1890  ?  What  was  the  in- 
crease in  population  from  1890  to  1900  ?  What  was  the 
per  cent  of  increase  in  the  ten  years  ? 

19.  A  newsboy  sold  25  papers  at  5^  each,  which  had 
cost  him  h^  each.  What  was  the  amount  of  his  profit? 
What  was  his  gain  per  cen<^  ^ 


188  PERCENTAGE 

235.  Oral  Exercises. 

1.  Find  25%,  50%,  and  75%  of  each  of  the  following: 
$100;  ISO;  |120;  40  A.;  36  in.;  2000  lb.;  16  oz.;  12 
mo.;  24  hr.;   360  da.;  144  sq.  in. 

2.  What  is  33i  %  and  6G|  %  of  each  of  the  following: 
$120?  $1200?  360  da.?  36  in.  ?  180  mi.?  27  ft.  ?  $1500? 
12  mo.  ?  60  min.  ?  24  yd.  ? 

3.  Find  121%,  371%,  62|  %,  and  871%  of  each  of  the 
following:  24  hr.;  $240;  $1;  20  mi.;  2000  1b.;  144  sq. 
in.;  360  da.;  $1200;  640  A.;  72  yd. ;  16  oz.;  216  cu.  in.; 
320  rd.;  $4000;  $.48. 

4.  Express  each  in  per  cent  :  ■^,  J,  |,  |,  f ,  J,  f ,  ^^,  |, 

h  h  h  h  h  h  h  h  h  h  ^^  H^  H'  If'  H^  21,  H,  If,  2, 8, 

10. 

5.  Express  each  as  a  common  fraction  in  lowest  terms: 
80%,  50%,  33J%,  25%,  125%,  20%,  120  %,  70%,  40  %, 
150%,  66|%,  14f  %,  75%,  133J-%,  175%,  180%,  12J%, 
140%,  160%,  112J%,  16|%,  90%,  37^%,  87J%,  62J%, 
1371%,  187J%,  110%,  130%. 

6.  Find  125%,  150%,  175%,  112J%,  1371%,  162J%, 
and  1871%  of  each  of  the  following:  24  hr.;  320  rd.; 
640  A.;  360  da,;  16  oz.;   20001b.;  $1200;  $4000;   $80. 

7.  Find  133J%,  120%,  166f  %,  140%,  and  160%  of 
each  of  the  following  :  $150;  30  da.;  360  da.;  $6000; 
120  rd.;  $250;   60 ff;  $1.80. 

8.  Find  the  number  of  which  30  is  33^%;  60  is  25  %; 
20  is  40%;  36  is  66^%;  35  is  125%;  120  is  120%;  48  is 
37J%;  90  is  150%;  180isl2l%;  180isll2}%;  42  is 
175%;  50is200%;  24isl60%;  200is40%;  80ia66J%; 
15  is  166f  %;  24  is  4%;  30  is  5%;  18  is  6%  ;  45  is  9%. 
38  is  110%;   12  is  2%;   130  is  200%. 


PERCENTAGE  189 

236.   Written  Exercises. 

1.  A  sum  of  money  borrowed,  together  with  the  in- 
terest on  it  for  one  year  at  7%,  amounted  to  $909.50. 
This  was  what  per  cent  of  the  money  borrowed?  Find 
the  sum  borrowed. 

2.  A  boy  spelled  correctly  45  words  in  a  test  of  50 
words.  What  per  cent  should  he  receive  as  his  standing 
in  the  test  ? 

3.  A  girl  missed  4  problems  in  an  arithmetic  test  con- 
taining 10  problems.  What  per  cent  of  the  problems  did 
she  miss  ?     What  per  cent  did  she  have  correct  ? 

4.  5  %  of  a  certain  amount  is  $20.     Find  the  amount. 

5.  Find  the  amount  when  8  %  of  the  amount  is  $240  ; 
$80. 

6.  A  farmer  had  24  cows  and  sold  16  of  them.  What 
per  cent  of  the  cows  did  he  sell  ?  What  per  cent  did  he 
have  left  ? 

7.  A  house  and  lot  was  advertised  for  sale  for  $8000. 
This  property  was  rented  for  $32.50  per  month.  The 
rent  amounted  to  what  per  cent  of  the  price  asked  for  the 
property  ? 

8.  If  a  man  spent  60%  of  his  savings  in  building  a 
barn  and  had  $400  left,  how  much  had  he  saved  ? 

9.  A  liveryman  made  40  %  on  the  cost  of  a  horse 
by  selling  it  at  a  profit  of  $36.     Find  the  cost  of  the  horse. 

10.  An  article  that  cost  a  retail  merchant  $  14  was  sold 
to  a  customer  at  a  profit  of  14|  %.  How  much  did  the 
customer  pay  for  the  article  ? 

11.  The  total  enrollment  in  a  certain  school  was  180 
pupils.  On  a  certain  day  150  pupils  were  present.  The 
number  present  was  what  per  cent  of  the  enrollment  ? 


190  PERCENTAGE 

237.   Oral  Exercises. 

1.  By  selling  a  horse  for  20  %  more  than  it  cost  him  a 
liveryman  gained  130.  How  much  did  the  horse  cost  him? 
For  how  much  did  he  sell  it  ? 

2.  What  is  2|  o/o  of  $400  ?  31  %  of  $60  ?  5  %  of  $1400? 
6%  of  $250?  10%  of  2000  lb.?  5^%  of  $200?  7%  of 
$150?   8%  of  $2500? 

3.  What  is  the  difference  between  1  %  and  .1  %  ?  be- 
tween ^  of  a  number  and  |  %  of  a  number  ? 

4.  What  is  \  %  of  $200  ?  .2  %  of  $400  ?  ^  %  of  $8000  ? 

5.  8  is  what  part  of  24  ?  8  is  what  per  cent  of  24  ? 
20  is  what  per  cent  of  25  ?  $20  is  what  per  cent  of  $30  ? 
$40  is  what  per  cent  of  $30  ? 

6.  A  boy  missed  1  word  in  a  spelling  lesson  of  20 
words.  At  the  same  rate,  how  many  would  he  have 
missed  in  a  lesson  of  100  words  ? 

7.  After  having  his  salary  raised  $10  a  month,  a 
clerk's  yearly  salary  amounted  to  $1620.  What  was  his 
monthly  salary  before  receiving  the  increase  ? 

8.  A  carriage  that  cost  $120  was  sold  for  $80.  The 
sale  price  was  what  per  cent  of  the  cost  ? 

9.  A  man's  monthly  salary  was  raised  from  $60  to  $75. 
What  per  cent  was  his  salary  increased  ? 

10.  25^  is  what  per  cent  of  30^?  The  cost  of  3  bars  of 
soap  when  bought  at  3  bars  for  25^  is  what  per  cent  of 
the  cost  when  bought  at  10^  a  bar  ? 

11.  What  per  cent  of  profit  is  made  when  articles  are 
bought  at  40^  a  dozen  and  sold  at  5^  apiece? 

12.  What  per  cent  of  profit  is  made  when  articles  are 
bought  at  10^  a  dozen  and  sold  at  2  for  5^? 


PEROENTAQE  191 

238.  Oral  Exercises. 

1.  If  I  of  the  value  of  a  piece  of  property  is  $1500, 
what  is  the  value  of  the  property  ? 

2.  If  I  of  a  man's  yearly  salary  is  f  1200,  what  is  his 
yearly  salary  ? 

3.  A  clerk  saved  $40  a  month,  which  was  |  of  his 
monthly  salary.     What  was  his  monthly  salary  ? 

4.  What  part  of  his  income  does  a  man  save  who  saves 
$60  a  month  from  an  income  of  $1200  a  year  ? 

5.  Frank  has  a  certain  sum  of  money  and  James  has 
I  as  much.  They  both  together  have  60^.  How  much 
money  has  each? 

The  moDey  of  both  together  is  how  many  thirds  of  Frank's  money  ? 

6.  Two  boys  took  a  piece  of  work  to  do  for  $6. 
One  boy  worked  twice  as  many  hours  as  the  other  boy. 
How  much  should  each  receive  ? 

7.  A  man  gave  Henry  $3  as  many  times  as  he  gave 
Walter  $4.  He  gave  $14  to  the  two  boys.  How  much 
did  each  receive  ? 

8.  Separate  $45  into  two  amounts  in  the  ratio  of  5 
to  4  ;    $36  into  three  parts  in  the  ratio  of  2,  3,  and  4. 

9.  In  a  school  of  120  pupils  there  were  5  girls  to 
every  3  boys.     Find  the  number  of  boys  and  girls. 

10.  Rob,  Fred,  and  Ada  together  received  $2.40  from 
their  father.  For  every  15^  that  Rob  received  Fred  re- 
ceived 10^,  and  Ada  5^.     How  much  did  each  receive  ? 

11.  A  newsboy  wished  to  make  an  estimate  of  his 
yearly  earnings,  so  he  kept  account  of  his  earnings  for  3 
weeks  and  found  that  he  earned  $6  the  first  week,  $4  the 
second  week,  and  $5  the  third  week.  At  the  same  rate, 
how  much  would  he  earn  in  a  year  ? 


192  PERCENTAGE 

2f39.  Commission.* 

A  person  who  transacts  business  for  another  frequently 
receives  as  his  pay  a  certain  rate  per  cent  of  the  amount 
involved  in  the  transaction.  This  is  known  as  his  com- 
mission. One  who  buys  or  sells  for  another  on  commis- 
sion is  called  a  commission  merchant,  a  broker,  or  an  agent. 

240.  Written  Exercises. 

1.  Find  2%  of  12400. 

2.  A  commission  merchant  sold  $2400  worth  of  hay 
for  a  farmer  and  charged  2  %  for  his  services.  Find  the 
amount  of  his  commission.  How  much  should  he  remit 
to  the  farmer,  after  deducting  his  commission  and  $300 
for  freight  charges  and  $150  for  cartage  ? 

3.  An  agent  received  $16  as  his  commission  for  sell- 
ing a  bill  of  goods  at  a  commission  of  5%.  Find  the 
amount  of  his  sales. 

4.  A  farmer  shipped  40  sacks  of  potatoes  to  a  commis- 
sion merchant,  who  sold  them  at  95^  a  sack.  After  de- 
ducting his  commission  of  5%,  how  much  should  he  remit 
to  the  farmer  ? 

5.  A  merchant's  profits  averaged  15%.  His  total 
sales  for  January,  1906,  amounted  to  $13,800.  Find  the 
cost  of  the  goods  sold.     Find  the  profits  for  the  month. 

6.  A  farmer  shipped  18  tons  of  hay  to  a  commission 
merchant,  who  sold  it  at  $9.50  per  ton.  How  much  did 
the  merchant  remit  to  the  farmer,  after  deducting  his  com- 
mission of  5  %  and  freight  and  cartage  charges  amounting 
to  $1.75  per  ton? 

*For  a  more  extended  treatment  of  Commission,  see  Appendix, 
pp.  262-264. 


PERCENTAGE  193 

7.  Find  the  net  proceeds  of  the  sale  of  860  lb.  of 
butter  at  18^  per  pound,  commission  6%. 

8.  A  real  estate  agent  received  a  commission  of  5% 
for  selling  a  city  lot.  Find  the  sale  price  of  the  lot,  if 
the  agent's  commission  amounted  to  $62.50. 

9.  If  the  salary  of  a  traveling  salesman  is  $20  a  week 
and  a  commission  of  l^  %  on  the  amount  of  his  sales,  how 
much  does  he  earn  in  a  week  in  which  his  sales  amount  to 
i  2254. 75? 

10.  A  carriage  dealer  offered  to  sell  a  certain  carriage 
for  $250  on  two  months'  time,  or  to  allow  a  discount  of 
2  %  for  cash.     Find  the  cash  price  of  the  carriage. 

11.  If  a  collector  retains  10  %  of  the  amount  of  a  cer- 
tain bill  for  collecting  it,  what  per  cent  of  the  amount  of 
the  bill  does  the  creditor  receive  ?  A  collector  remitted 
to  a  creditor  $126  as  the  net  proceeds  of  a  collection,  after 
retaining  his  commission  of  10%.  Find  the  amount  of 
the  bill  collected. 

12.  After  deducting  his  commission  of  4%,  an  agent 
remitted  $79.20  to  a  shipper.  Find  the  amount  of  the 
sales. 

13.  The  amount  received  by  a  shipper,  after  a  commis- 
sion of  5  %  has  been  deducted,  is  what  per  cent  of  the 
amount  of  the  sales?  A  shipper  received  $60.80  as  the 
net  returns  of  a  sale  of  some  potatoes,  after  paying  a  com- 
mission of  5  % .     Find  the  amount  of  the  sale. 

14.  A  dairyman  shipped  1250  lb.  of  butter  to  a  com- 
mission merchant,  who  sold  it  at  22^  per  pound.  If  the 
cost  of  shipping  was  $2.40  and  the  cartage  amounted  to 
$1.75,  how  much  did  the  shipment  net  the  dairyman,  after 
paying  a  commission  of  4  %  ? 

AB.  — IS 


194  PERCENTAGE 

241.  Oral  Exercises. 

1.  If  a  boy  sells  1  newspaper  for  what  2  papers  cost 
him,  what  per  cent  of  profit  does  he  make? 

2.  If  a  baker  sells  2  pies  for  what  3  pies  cost  him, 
what  per  cent  of  profit  does  he  make? 

3.  A  merchant  sold  5  yd.  of  cloth  for  what  6  yd.  cost 
him.     What  per  cent  of  profit  did  he  make? 

4.  What  per  cent  of  profit  does  a  grocer  make  who 
buys  canned  tomatoes  at  the  rate  of  3  cans  for  25^  and 
sells  them  at  the  rate  of  2  cans  for  25/? 

5.  A  dealer  marked  his  goods  so  that  he  would  make 
30  %  profit  on  them.  In  order  to  dispose  of  his  goods,  he 
was  obliged  to  sell  them  at  a  discount  of  10  %  from  the 
marked  price.     What  per  cent  of  profit  did  he  make? 

6.  A  collector  was  allowed  a  commission  of  20%  on 
a  bill  of  $80.     What  amount  did  the  creditor  receive  ? 

7.  A  dealer  marked  his  goods  at  20  %  above  cost.  In 
order  to  close  out  his  stock,  he  was  obliged  to  sell  the 
goods  at  a  discount  of  25%.  Did  he  gain  or  lose,  and 
what  per  cent? 

8.  There  are  4  boys  and  8  girls  in  a  class  in  arithmetic. 
What  per  cent  of  the  pupils  in  the  class  are  girls? 

9.  The  enrollment  of  pupils  in  a  class  was  25  in  1905 
and  30  in  1906.     What  was  the  per  cent  of  increase  ? 

10.  On  a  certain  day  a  boy  missed  3  words  out  of  15  in 
spelling.  What  was  the  per  cent  of  words  correctly 
spelled? 

11.  An  agent  received  a  commission  of  5  %  for  selling 
a  lot  for  $1500.     Find  the  amount  of  his  commission. 

12.  An  agent's  commission  of  5%  for  selling  a  city  lot 
amounted  to  $60.     For  what  amount  did  he  sell  the  lot? 


INSURANCE  195 

242.  Insurance.*  i.  Owners  of  buildings,  merchandise, 
etc.,  generally  protect  themselves  against  loss  by  fire  by 
having  such  property  insured.  Insurance  of  property 
against  loss  by  fire  is  called  fire  insurance,  against  loss  by 
sea  marine  insurance.  What  is  life  insurance?  accident 
insurance?     Name  other  forms  of  insurance. 

2.  The  written  agreement  between  the  insurance  com- 
pany and  the  person  protected  is  called  a  policy.  Examine 
a  fire  insurance  policy.  The  amount  paid  for  insurance  is 
called  the  premium.  The  rates  of  insurance  are  expressed 
as  a  rate  per  cent  on  the  face  of  the  policy,  or  as  a  speci- 
fied sum  for  each  |100,  or  for  each  11000,  of  the  face  of 
the  policy. 

243.  Written  Exercises. 

1.  Mr.  Wilson  insured  his  store  for  $6000.  The  in- 
surance cost  him  1J%.     Find  the  premium. 

2.  Mrs.  Hardy  insured  her  house,  valued  at  $8000,  for 
I  of  its  value.  Find  the  amount  of  the  face  of  the  policy. 
The  insurance  cost  her  $1.40  on  each  $100  and  extended 
for  three  years.     How  much  did  the  insurance  cost  her? 

3.  If  90  %  of  a  sum  is  $28.80,  what  is  the  sum  ? 

4.  For  what  price  was  a  city  lot  sold  if  the  agent's 
commission  of  5%  amounted  to  $87.50?  How  much  did 
the  owner  receive  ? 

Find  the  premium  on  the  following  amounts  of  in- 
surance at  the  rates  given  : 

5.  $4000  at  1\%.  8.    $5600  at  $1.20  per  $100. 

6.  $2400  at  If  %.  9.    $1400  at  $1.35  per  $100. 

7.  $12,000  at  11%.  10.    $4250  at  $1.80  per  $100. 

*  For  a  more  extended  treatment  of  Insurance,  see  Appendix, 
pp.  278-283. 


196  PERCENTAGE 

11.  Mr.  Rogers  built  a  house  that  cost  him  $4500.  It 
cost  him  11800  additional  to  furnish  it.  To  protect  him- 
self against  the  complete  loss  of  his  property  by  fire,  he 
insured  his  house  for  $3000  and  his  household  goods  for 
$1200.  The  insurance  for  three  years  cost  him  !{%  of 
the  face  of  the  policy. 

a.   Find  the  cost  of  the  insurance. 

h.  If  the  house  and  contents  were  destroyed  by  fire, 
how  much  insurance  would  he  receive  ? 

c.  What  would  be  the  amount  of  his  loss,  including 
the  amount  paid  for  insurance  ? 

d.  If  the  house  were  damaged  to  the  extent  of  $400, 
how  much  would  he  receive? 

12.  Two  men  own  a  store  in  partnership.  One  has 
$16,000  invested  in  it,  and  the  other  has  $10,000.  What 
part  of  the  store  does  each  own?  If  the  store  were  sold 
for  $39,000,  what  part  of  this  amount  would  each  re- 
ceive? How  much  would  each  receive?  If  the  store 
were  damaged  by  fire  to  the  extent  of  $13,000,  how 
much  would  each  lose? 

13.  A  hotel  valued  at  $80,000  was  insured  for  $50,000 
in  one  company  and  for  $25,000  in  a  second  company. 
How  much  would  each  company  be  liable  for  (a)  if  the 
hotel  were  totally  destroyed ;  (h)  if  it  were  damaged  to 
the  extent  of  $12,000?  of  $30,000? 

14.  What  was  the  amount  of  commission  received  by  an 
architect  who  charged  a  commission  of  5  %  for  drawing 
the  plans  and  supervising  the  construction  of  a  house  that 
cost  $4500,  exclusive  of  the  architect's  fees? 

15.  Write  five  insurance  problems  based  on  conditions 
in  your  community. 


PERCENTAGE  197 

244.  Oral  Exercises. 

Express  the  part  and  the  per  cent  the  first  quantity  is  of 
the  second : 

1.  130,  140.  6.  12.50,  $3.  11.  11200, 11500. 

2.  $40, 150.  7.  80  A.,  160  A.  12.  45  T.,  60  T. 

3.  20  mi.,  25  mi.       8.  10  yd.,  16  yd.  13.  80  A.,  320  A. 

4.  40  ft.,  60  ft.        9.  $4, 124.  14.  2000  ft.,  2200  ft. 

5.  1100,1120,       10.  60  lb.,  100  lb.  15.  |12,  |200. 

16.  Express  the  ratio  of  the  second  quantity  to  the  first 
in  each  of  the  above  in  the  form  of  a  fraction  in  lowest 
terms  and  in  per  cent. 

245.  Oral  Exercises. 

1.  A  collector's  commission  of.  5%  amounted  to  $30. 
Find  the  amount  of  the  bill  collected. 

2.  After  deducting  his  commission  of  20%,  a  collector 
remitted  f  24  to  the  creditor.  Find  the  amount  of  the 
bill  collected. 

3.  Mr.  Wright  has  $4500  out  on  interest  at  6  %.  His 
annual  taxes  on  the  money  are  $20.  What  is  his  net 
annual  income  from  the  $4500? 

4.  The  yield  from  a  certain  field  was  30  bu.  of  oats  to 
the  acre  in  1904  and  40  bu.  to  the  acre  in  1906.  What 
was  the  per  cent  of  increase  in  the  yield  in  1906  over  the 
yield  in  1904? 

5.  The  enrollment  in  a  certain  school  in  1906  was  36 
pupils,  which  was  an  increase  of  20  %  over  1905.  What 
was  the  number  of  pupils  enrolled  in  1905  ? 

6.  40%  of  the  pupils  in  a  certain  school  are  boys. 
There  are  24  girls  in  the  school.  How  many  pupils  are 
there  in  the  school? 


198  PERCENTAGE 

246.  Taxes.  *  l.  What  are  some  of  the  expenses  of  a  city 
government?  of  a  state  government?  of  the  national  gov- 
ernment? The  money  necessary  for  the  maintenance  of 
state  and  local  governments  is  derived  mainly  from  taxes 
levied  upon  persons,  property,  and  business. 

All  movable  property,  such  as  household  goods,  money,  cattle, 
ships,  etc.,  is  called  personal  property.  Immovable  property,  such  as 
lands,  buildings,  mines,  etc.,  is  called  real  estate,  or  real  property. 
Both  forms  of  property  are  subject  to  taxation. 

2.  For  the  purpose  of  taxation,  the  value  of  all  taxable 
property  is  estimated  by  a  public  officer  called  an  assessor. 
Property  is  not  generally  assessed  at  its  full  value. 

3.  The  rate  of  taxation  is  expressed  as  a  per  cent  on  the 
assessed  valuation,  or  as  a  specified  sum  on  each  $1,  or  on 
each  ^100,  of  assessed  valuation.  Thus,  a  tax  rate  of  1^  % 
may  be  stated  as  a  tax  of  1|^  (on  each  |1),  or  of  $1.50 
(on  each  $100). 

4.  The  national  government  is  supported  mainly  by  rev- 
enues derived  from  taxes  levied  upon  goods  imported  from 
other  countries,  called  duties,  or  customs,  and  from  internal 
revenues,  which  consist  chiefly  of  taxes  levied  upon  the 
manufacture  of  liquors  and  tobacco  products. 

Some  imports  are  admitted  without  duty.  These  are  said  to  be  on 
the  free  list.  Nearly  all  imports  are  subject  either  to  an  ad  valorem 
or  a  specific  duty^  or  both. 

5.  An  ad  valorem  duty  is  a  tax  of  a  certain  rate  per  cent 
on  the  cost  of  the  goods. 

6.  A  specific  duty  is  a  tax  of  a  specified  amount  per  pound, 
yard,  etc.,  without  reference  to  the  cost  of  the  goods. 

7.  Customhouses  have  been  established  at  all  ports  wliere 
vessels  are  authorized  to  land  cargoes.  The  revenues  are 
collected  by  federal  officers  stationed  at  ports  of  entry. 

*  For  a  more  extended  discussion  of  Taxes,  see  Appendix,  pp.  269-273. 


TAXES  199 

247.  Written  Exercises. 

1.  A  man  had  16000.  He  invested  11500  in  a  city 
Ipt.     What  per  cent  of  his  money  did  he  invest  ? 

2.  A  certain  city  had  an  assessed  valuation  of 
#8,000,000.  The  amount  needed  to  defray  the  expenses 
of  the  city  for  a  year  was  estimated  at  1100,000.  The 
amount  needed  for  expenses  was  what  per  cent  of  the 
assessed  valuation? 

3.  The  assessed  valuation  of  a  certain  city  is  $  12,000,000 
and  the  amount  to  be  raised  by  taxation  is  1180,000.  What 
rate  of  taxation  is  necessary  in  order  to  raise  this  amount? 

4.  What  is  the  amount  of  an  agent's  commission  for 
selling  real  estate  for  $150,000  at  a  commission  of  1^  %  ? 

5.  What  is  the  amount  of  a  man's  taxes  on  property 
assessed  at  16000  if  the  tax  rate  is  11.20  on  each  1 100? 

6.  A  real  estate  agent  received  #84  for  selling  a  piece 
of  property  at  a  commission  of  2%.  Find  the  selling 
price  of  the  property. 

7.  The  assessed  valuation  of  a  certain  farm  is  -$3600. 
This  is  40  %  less  than  the  actual  value  of  the  farm.  Find 
the  value  of  the  farm. 

8.  What  per  cent  on  his  investment  did  a  boy  make 
who  bought  a  pony  for  $40  and  sold  him  for  $50? 

9.  The  assessed  valuation  of  the  property  in  a  county 
'm  $42,000,000,  and  $672,000  is  to  be  raised  by  taxation. 
Express  the  rate  of  taxation  in  three  ways. 

10.    Find  the  rate  of  taxation  on: 

a.  Valuation,  $450,000;  taxes,  $6000. 
h.  Valuation,  $275,000;  taxes,  $2475. 
c.   Valuation,  $360,000;  taxes,  $6300. 


200  PERCENTAGE 

11.  What  rate  of  commission  was  charged  by  a  col- 
lector who  charged  $15  for  collecting  a  debt  of  $225  ? 

12.  The  premium  on  an  insurance  of  $4500  is  $60. 
What  is  the  rate  of  premium  ? 

13.  The  premium  received  for  insuring  a  store  at  IJ  % 
was  $105.     What  was  the  amount  of  insurance? 

14.  At  the  rate  of  1|  %,  how  much  is  the  tax  on  prop- 
erty assessed  at  $4500  ? 

15.  When  the  valuation  and  the  rate  of  taxation  are 
given,  how  may  the  tax  be  found  ?     Find  the  tax  on : 

a,    $12,000  at  1|%;  at  .8%  ;  at  1.4%;  atlf%. 
h.    $10,000  at  $1.20  per  $100;  at  $.80  per  $100. 

c.  $6000  at  8  mills  on  a  dollar  ;  at  7.6  mills  on  a  dollar. 

d.  $3600  at  $.007  on  a  dollar  ;  at  $.014  on  a  dollar. 

16.  If  a  broker  received  a  commission  of  IJ  %  for  his 
services,  find  the  amount  of  his  brokerage  for  buying 
2450  cwt.  of  wheat  at  $1.34  per  cwt.  If  this  wheat  was 
bought  for  a  milling  company,  what  was  the  total  cost  of 
♦^^he  wheat  to  the  company  ?  the  cost  per  cwt.  ? 

17.  If  a  traveling  salesman  sells  on  an  average  $400 
worth  of  goods  every  week,  which  of  the  following  offers 
should  he  accept  from  the  wholesale  firm  :  (a)  a  salary 
of  $25  per  week  and  expenses;  (5)  a  salary  of  $15  a 
week  and  expenses,  and  a  commission  of  5  %  on  all  sales 
over  $300  per  week;  (c)  or  his  expenses  and  a  commis- 
sion of  8  %  on  all  sales  ? 

18.  When  the  tax  and  the  rate  of  taxation  are  given, 
how  may  the  valuation  be  found?     Find  the  valuation: 

a.    Tax,  $60;  rate  of  taxation,  1J%. 

h.    Tax,  $120 ;  rate  of  taxation,  8  mills  on  a  dollar. 

(?.    Tax,  $96 ;  rate  of  taxation,  $1.20  per  $100. 


INSURANCE  AND  TAXES  201 

19.  What  was  the  amount  of  insurance  if  the  premium 
received  for  insuring  a  house  at  $1.40  per  $100  was  $49  ? 

20.  Furniture  yalued  at  $600  was  insured  for  $400. 
For  what  part  of  its  value  was  the  furniture  insured? 
The  premium  paid  for  3  years  was  $8.  What  was  the 
rate  of  premium  paid  ? 

21.  The  pupils  of  the  advanced  arithmetic  class  in  a 
certain  school  were  told  that  the  school  building  was 
insured  for  |  of  its  estimated  value,  and  that  the  annual 
premium  at  1  %  was  $75.  They  were  asked  to  find  the 
estimated  value  of  the  building.  One  pupil  found  the 
value  to  be  $5625.     Was  his  answer  correct  ? 

22.  A  certain  school  district  voted  $12,000  to  erect  a 
new  schoolhouse.  The  assessed  valuation  of  the  property 
in  the  district  was  $600,000.     Find  the  rate  of  taxation. 

23.  If  a  tax  collector  in  a  certain  city  receives  a  com- 
mission of  2  %  for  collecting  taxes,  what  per  cent  of  the 
amount  collected  does  the  city  receive  ?  Find  the  amount 
of  taxes  that  must  be  levied  in  order  that  a  city  may 
receive  $19,600,  after  allowing  a  collector  a  commission 
of  2%  for  collecting. 

119,600  =  98%  of  the  sum  levied. 

24.  Property  worth  $9000  was  assessed  at  $6000.  The 
rate  was  $1.50  for  each  $100  of  assessed  valuation.  Had 
this  property  been  assessed  at  its  full  value,  what  rate  of 
taxation  would  have  yielded  the  same  amount  of  taxes  ? 

25.  Examine  a  tax  receipt.  Is  a  separate  entry  made 
for  taxes  on  personal  property  and  on  real  property? 
Is  there  an  entry  for  school  taxes  ? 

26.  Make  and  solve  five  problems  in  taxes,  using  when 
possible  the  actual  rates  in  your  county  or  city. 


202  PERCENTAGE 

248.  Customs  and  Duties.  The  following  rates  of  cus- 
toms are  from  the  schedule  adopted  by  Congress  in  1897, 
commonly  known  as  the  Dingley  Tariff : 


Newspapers,  periodicals,  free. 

Hay,  %  4  per  ton. 

Coffee,  free. 

Carpets  (velvet),  60^  per  sq.  yd. 

Musical     instruments,     45  % 

and  40%  ad.  val. 

ad.  val. 

Table  knives,  16^  each  and  15% 

Potatoes,  25  f  per  bu. 

ad.  val. 

Tea,  free. 

Paintings,  20%  ad.  val. 

249.    Written  Exercises. 

1.  Find  the  duty  on  60  sq.  yd.  of  velvet  carpet  worth 
$1.50  per  square  yard. 

2.  What  is  the  duty  on  45  tons  of  hay  ? 

3.  What  is  the  duty  on  a  violin  worth  180  ? 

4.  A  painting  valued  at  12500  was  purchased  in  Italy 
and  brought  to  the  United  States.  Find  the  amount  of 
customs  on  it. 

5.  Find  the  amount  of  the  duty  on  6  doz.  table  knives 
worth  $1.80  per  dozen. 

6.  Why  are  tea  and  coffee  on  the  free  list,  while  a  duty 
of  25^  per  bushel  is  placed  upon  potatoes  ? 

7.  What  were  the  net  proceeds  of  an  auction  sale,  if  the 
sales  amounted  to  11215.40,  and  the  auctioneer  received 
a  commission  of  10  %  ? 

8.  After  deducting  his  commission  of  5%  and  $12.50 
for  freight  and  cartage,  a  commission  merchant  remitted 
$633.50  to  the  shipper.     Find  the  amount  of  the  sales. 

9.  A  city  lot  that  cost  $1600  was  sold  for  $1800. 
Find  the  gain  per  cent. 


PERCENTAGE  203 

250,  Oral  Exercises. 

1.  What  is  the  price  of  coal  a  ton  when  it  is  selling  at 
i.25  a  hundredweight  ? 

2.  When  hay  is  selling  at  $12  a  ton,  what  is  its  price 
per  hundredweight  ? 

3.  If  I  of  the  length  of  a  certain  bridge  is  240  ft., 
how  long  is  the  bridge  ? 

4.  If  the  interest  for  one  year  at  5%  is  $80,  what  is 
the  sum  on  which  the  interest  is  paid  ? 

5.  A  boy  shot  10  times  at  a  target  and  hit  it  8  times. 
Express  as  per  cent  the  ratio  of  the  number  of  accurate 
shots  to  the  number  of  shots  taken. 

6.  On  a  certain  day  a  girl  missed  3  out  of  12  words  in 
a  spelling  lesson.  What  per  cent  of  the  words  did  she 
spell  correctly  ? 

7.  A  baseball  team  played  8  games  and  lost  3  of  them. 
What  per  cent  of  the  games  played  did  the  team  win  ? 

8.  A  girl  was  absent  from  school  4  days  and  present 
16  days  during  a  school  month.  What  per  cent  of  the 
time  was  she  present  ? 

9.  A  man  paid  a  tax  of  1J%  on  property  valued  at 
$4000.     Find  the  amount  of  his  tax. 

10.  A  commission  merchant  received  $20  for  selling 
$1000  worth  of  produce.  What  was  his  rate  of  com- 
mission ? 

11.  If  a  spelling  lesson  consists  of  25  words,  what  per 
cent  of  the  lesson  is  each  word  ?  What  per  cent  of  the 
words  does  a  boy  spell  correctly  who  misspells  4  words  ? 

12.  A  boy  caught  a  ball  6  times  and  missed  it  2  times. 
The  number  of  times  he  caught  the  ball  is  what  per  cent 
of  the  number  of  chances  he  had  to  catch  it  ? 


204  PERCENTAGE 

251.  Trade  Discount.*  i.  Manufacturers  and  wholesale 
dealers  issue  catalogues  describing  articles  sold  by  them 
and  giving  their  list  prices,  A  discount  from  the  list 
price  is  made  to  retail  dealers  and  sometimes  to  other 
customers,  particularly  when  goods  are  purchased  in  large 
quantities.  Such  a  discount  is  generally  known  as  trade 
discount,  or  commercial  discount. 

2.  Several  successive  discounts  are  sometimes  allowed. 
Thus,  an  article  may  be  sold  subject  to  discounts  of  25  %, 
10  %,  and  5  %  ;  that  is,  a  discount  of  25  %  is  made  from 
the  list  price,  and  a  second  discount  of  10  %  is  made  from 
the  price  after  making  the  discount  of  25  %,  and  a  third 
discount  of  5  %  is  made  on  the  price  after  the  two  dis- 
counts have  been  made.  A  separate  cash  discount  is 
usually  allowed  when  payment  is  made  within  a  specified 
time  after  the  purchase  of  the  goods. 

252.  Written  Exercises. 

1.  Find  the  net  cash  price  to  a  retail  hardware  mer- 
chant of  a  stove  listed  at  $45,  trade  discounts  of  20  % 
and  10  (fo^  and  a  cash  discount  of  5  %. 

Model  :      $  45,        list  price.  ^^^  ^'^^  discount  is  20% 

^  n     .   1.  .  of  $45,  or  $9.     The  price 

_9,        first  discount.        ^^^^^  ^^^^.^^  ^^^^  ^.^^^^^^ 

%  36,        second  price.         is  $  45  -  $  9,  or  $  36.    The 

3.60,  second  discount,   second  discount  is  10%  of 

$32.40,  third  price.  ^36,  or  $3.G0.    The  price 

1.62,  cash  discount.       ^^^^  ^^^"^g   *^«   ««^^"^ 
discount     is    $  36  -  1 3.60, 


$30.78,  net  price.  "  d^on  <n      -n        xu-     ' 

'  ^  or    $32.40.      From  thw   a 

cash  discount  of  5  %  is  deducted,  leaving  the  net  price  $30.78. 

Instead  of   deducting  each  discount  separately,  the  sum   of  the 

several  discounts  may  be  stated  as  a  single  discount.     This  may  be 

*  For  a  more  extended  discussion  of  Trade  Discount,  see  Appendix,  p.  865. 


TRADE   DISCOUNT  206 

found  thus :  A  discount  of  20  %  from,  the  list  price  reduces  the  cost 
to  80%  of  the  list  price;  and  the  second  discount  reduces  it  to  90% 
of  this,  or  to  90%  of  80%  of  the  list  price,  which  is  72  %  of  the  list 
price,  and  the  cash  discount  reduces  it  to  95%  of  72%  of  the  list 
price,  or  to  68.4  %  (68|  %)  of  the  list  price  *  Compare  68|  %  of  $  45 
with  the  answer  found. 

In  figuring  discounts,  use  the  shortest  method  in  every 
part  of  the  problem. 

2.  Which  is  the  greater  discount,  a  single  discount  of 
25  %  or  a  discount  of  20  %  and  a  second  discount  of  5  %  ? 

3.  If  the  hardware  merchant  (Prob.  1)  sold  the  stove 
for  $45,  how  much  was  his  profit  if  he  paid  out  $  2.50  for 
freight  and  cartage  ?     What  per  cent  profit  did  he  make  ? 

4.  What  was  the  net  cash  price  to  a  jeweler  of  a  watch 
listed  at  $35,  discounts  30  %,  15  %,  5  %,  and  a  cash  dis- 
count of  2  %  ?  If  the  jeweler  sold  the  watch  for  $  35, 
what  per  cent  profit  did  he  make  on  it  ? 

5.  A  piano  firm  bought  a  piano  listed  at  $350  and 
received  discounts  of  40  %,  20  %,  and  10  %.  How  much 
did  the  firm  make  on  the  piano  by  selling  it  at  $350? 
What  per  cent  profit  was  made  by  the  firm  ? 

6.  How  much  does  a  dealer  make  on  a  carriage  listed 
at  $120,  if  he  buys  it  at  a  discount  of  20  %,  5  %,  and 
takes  advantage  of  a  cash  discount  of  2  %,  and  sells  it 
at  the  list  price  ? 

7.  How  much  less  does  a  dealer  pay  for  a  wagon  listed 
at  $150,  if  he  is  allowed  a  single  discount  of  35  %  than  if 
he  is  allowed  successive  discounts  of  15  %,  15  %,  and  5  %  ? 

8.  What  per  cent  of  $48  is  $60  ?  of  $50  is  $30  ?  of  $120 
is  $100  ?  of  $100  is  $120  ?  of  $150  is  $200  ? 

*  The  single  equivalent  discount  may  be  found  by  adding  together 
20%,  10%of  80%,  or8%,and5%of  72%,  or  3.6%.  20%+8%+3.6%  = 
31.6  %. 


206  PERCENTAGE 

253.   Oral  Exercises. 

1.  A  man  bought  a  house  for  f  4000  and  sold  it  at  a 
gain  of  20  %.     For  how  much  did  he  sell  it  ? 

2.  A  farmer  sold  200  sacks  of  potatoes,  which  was  80  % 
of  his  entire  crop.     How  much  was  his  entire  crop  ? 

3.  If  the  price  of  steak  was  raised  from  12  ^  a  pound 
to  15  ^  a  pound,  what  was  the  per  cent  of  increase  in 
price  ? 

4.  The  total  enrollment  in  a  certain  school  was  40 
pupils,  and  the  average  number  in  daily  attendance  was 
35.  The  average  daily  attendance  was  what  per  cent  of 
the  enrollment  ? 

5.  An  agent  received  f  6  for  collecting  a  bill  of  $  30. 
What  was  his  per  cent  of  commission  ? 

6.  A  man  bought  a  farm  for  $4000  and  sold  it  for 
1 5400,  what  was  his  per  cent  of  gain  ? 

7.  At  what  price  must  a  dealer  sell  carriages  that  cost 
$  120  to  make  a  profit  of  33J  %  ?  20  %  ?  25  %  ?  12^  %  ? 
10%? 

8.  After  increasing  his  capital  by  $1200,  a  merchant 
had  $  4200  invested  in  his  business.  What  amount  had  he 
invested  before  increasing  his  capital  ?  By  what  per  cent 
of  itself  was  the  original  capital  increased  ? 

9.  By  selling  a  city  lot  for  $  1500,  a  man  gained  25  %. 
Find  the  cost  of  the  lot. 

10.  Property  worth  $  6000  was  assessed  for  purposes  of 
taxation  at  i  4000.  For  what  per  cent  of  its  value  was  the 
property  assessed  ? 

11.  What  monthly  rental  must  a  man  get  from  property 
valued  at  $3000  to  yield  a  net  income  of  6  %,  if  it  costs 
him  $  60  a  year  to  maintain  his  property  ? 


INTEREST  207 


INTEREST 


264.  1.  On  July  1, 1907,  Charles  H.  Thomas  borrowed 
of  Joseph  R.  White  i  300,  which  he  promised  to  return  in 
one  year,  with  interest  at  6  per  cent.  As  an  acknowledg- 
ment of  his  indebtedness  and  as  a  promise  to  pay,  Mr. 
Thomas  gave  Mr.  White  his  note,  of  which  the  following 
is  a  copy: 


$300.  fSe.\.k£.t£Af,  (^clL,  fioUf  f,  190/. 

€yL&  if&av after  date,  for  value  received,...  J... 

promise  to  pay  to  jao^eyfsA  R.  -W^kite^,  or  order, 

100 

zvith  interest  thereon  at  6%  per  annum  from  date  until 

paid. 

^/icuvt&Q^  /if.  ^kayyuao^. 


2.  The  sum  specified  in  a  note  is  called  its  face,  or  the 
principal. 

3.  What  is  the  principal  and  the  rate  of  interest  speci- 
fied in  the  above  note  ?  When  and  where  was  the  note 
executed  ?     On  what  date  did  the  above  note  fall  due  ? 

4.  The  person  to  whom,  or  to  whose  order,  the  amount 
named  in  a  note  is  to  be  paid  is  called  the  payee,  and  the 
person  by  whom  the  note  is  to  be  paid  is  called  the  payer. 
In  the  above  note  who  is  the  payer  and  who  the  payee  ? 
What  is  the  meaning  of  the  words  or  order?    per  annum? 

5.  Has  any  provision  been  made  in  the  above  note  for 
the  payment  of  interest  beyond  the  period  of  one  year  ? 


208  PERCENTAGB 

6.  Where  in  the  note  did  Mr.  Thomas  acknowledge 
that  he  had  received  of  Mr.  White  something  of  the  value 
of  $  300  ? 

7.  As  the  note  was  made  payable  to  Mr.  White,  or 
order^  it  is  said  to  be  negotiable ;  that  is,  it  may  be  passed 
from  one  person  to  another,  and  it  becomes  payable  to 
the  person  to  whom  it  is  ordered  paid.  Six  months 
after  the  note  was  executed,  Mr.  White  bought  a  city 
lot  of  J.  C.  Anderson,  and  as  part  payment  for  the  lot, 
he  transferred  the  note  to  Mr.  Anderson.  In  making 
the  transfer,  Mr.  White  wrote  on  the  back  of  the  note, 
over  his  own  signature,  "Pay  to  J.  C.  Anderson,  or 
order."  By  this  indorsement,  the  note  was  made  payable 
to  Mr.  Anderson. 

8.  Find  the  interest  on  the  note  to  July  1,  1908. 

9.  The  sum  of  the  principal  and  interest  is  called  the 
amount. 

10.    What  was  the  amount  of  the  note  on  July  1,  1908  ? 

255.  Method  of  Aliquot  Parts. 

1.  The  unit  of  time  for  which  interest  is  computed  is 
usually  one  year.  The  interest  on  a  given  note  for  three 
years  is  how  many  times  the  interest  for  one  year?  The 
interest  for  six  months  is  what  part  of  the  interest  for 
one  year? 

2.  What  part  of  the  interest  for  one  year  is  the  interest 
for  3  months?  for  2  months.?  for  4  months?  for  1  month  ^ 

3.  The  interest  for  1  month  is  what  part  of  the  in- 
terest for  6  months  ?  for  4  months  ?  for  3  months?  for  2 
months  ? 

4.  The  interest  for  5  months  is  the  interest  for  4 
months  plus  the  interest  for  what  part  of  4  months? 


INTEREST  209 

5.  Find  the  interest  on  1400  for  1  yr.  and  6  mo. at  7%. 
Model:  |400 

.07 
$28.00  =  interest  for  1  yr. 

4  mo.  =  J  yr.     9.33  =  interest  for  4  mo. 
1  mo.  =  J  of  4  mo.     2.33  =  interest  for  1  mo. 

139.66  =  interest  for  1  yr.  and  5  mo. 

Explanation.  First  find  the  interest  for  1  yr.  by  multiplying 
the  principal  by  .07.  Next  find  the  interest  for  4  mo.  by  dividing  the 
interest  for  1  yr.  by  3 ;  then  find  the  interest  for  1  mo.  by  dividing 
the  interest  for  4  mo.  by  4.  The  sum  of  these  three  amounts  is  the 
interest  for  1  yr.  and  5  mo.     In  dividing  drop  all  fractions  of  cents. 

6.  From  the  interest  for  one  year,  the  interest  for  any 
number  of  months  may  be  found  by  taking  the  following 

parts; 

1  mo.  =  ^  yr.  3  mo.  =  ^  yr. 

2  mo.  =  I  yr.  4  mo.  =  |  yr. 

6  mo.  =  I  yr. 

5  mo.  (4  mo.  and  1  mo.)  =  ^  yr.  plus  |  of  |  yr. 

7  mo.  (6  mo.  and  1  mo.)  =  ^  yr.  plus  |  of  |  yr. 

8  mo.  (4  mo.  and  4  mo.)  =  ^  yr.  plus  J  yr. 

9  mo.  (6  mo.  and  3  mo.)  =  |  yr.  plus  ^  of  ^  yr. 

10  mo.  (6  mo.  and  4  mo.)  =  I  yr.  plus  }  yr. 

11  mo.  (6  mo.  and  5  mo.)  =  I  yr.  plus  i  yr.  plus  |  of  i  yr. 
Into  what  other  suitable  parts  for  finding  interest  may  each  be 

divided  :  9  mo.  ?  8  mo.  ?  10  mo.  ? 

256.  Written  Exercises. 

Find  the  interest  and  the  amount  of  : 

1.  8250,  1  yr.  6mo.,  8%.  6.  1260,  1  yr.  10  mo.,  8%. 

2.  8560,  2  yr.  4  mo.,  6%.  7.  |720,  1  yr.  3  mo.,  5%. 

3.  8875,  3  yr.  5  mo.,  7%.  8.  81200,  10  mo.,  6%. 

4.  8100,  1  yr.  9  mo.,  7%.  9.  82400,  2  yr.  3  mo.,  6%. 

5.  8620,  7mo.,  4J%.  lo.  8500,  1  yr.'8  mo.,  4%. 

AB.  —  14 


210  PERCENTAGE 

257.  Interest  for  Years,  Months,  and  Days. 

1.  It  is  sometimes  necessary  to  find  the  interest  for 
years,  months,  and  days,  in  which  case  thirty  days  are 
usually  regarded  as  one  month. 

2.  When  the  interest  for  one  month  is  known,  how  may 
the  interest  be  found  for  15  da.  ?  for  10  da.  ?  for  6  da.  ? 
for  5  da.  ?  for  3  da.  ?  for  1  da.  ?  When  the  interest  for 
6  da.  is  known,  how  may  the  interest  be  found  for  1  da.  ? 

3.  Find  the  interest  on  |150  for  1  yr.  7  mo.  14  da.  at  8  %. 

Model:  $150 

m 

$12.00  =  interest  for  1  yr. 
6  rao.  =  I  yr.  6.00  =  interest  for  6  mo. 

1  mo.  =  ^  of  ^  yr.  1.00  =  interest  for  1  mo. 

10  da.  =  I  rao.  .33  =  interest  for  10  da. 

3  da.  =  fj  mo.  .10  =  interest  for  3  da. 

1  da.  =  I  of  3  da.  .03  =  interest  for  1  da. 

$19.46  =  interest  for  1  yr.  7  mo.  14  da. 

4.  From  the  interest  for  one  month,  the  interest  for  any 
number  of  days  may  be  found  as  in  the  following : 

22  da.  (10  da.  and  10  da.  and  2  da.)  =  |  mo.  plus  |  mo.  plus  ^  of  ^  mo. 

18  da.  (6  da.  and  6  da.  and  6  da.)  =  \  mo.  taken  3  times.  Is  this 
easier  than  to  separate  18  da.  into  the  parts  15  da.  and  3  da.,  or 
into  the  parts  10  da.  and  6  da.  and  2  da.  ?    Explain  and  illustrate. 

5.  Determine  for  each  number  of  days,  from  1  to  29, 
how  the  interest  can  be  found  most  readily,  when  the  in- 
terest for  one  month  is  known.  Compare  your  results 
with  those  determined  by  other  pupils,  to  see  who  has  the 
best  method.  Test  each  method  by  taking  some  amount 
as  the  interest  for  one  month. 

6.  Write  a  note,  naming  some  pupil  as  payee  and  your- 
self as  the  maker,  and  find  the  amount  of  the  note  for 
1  yr.  5  mo.  12  da. 


INTEREST  211 

258.  Written  Exercises. 

Find  the  interest  on: 

1.  $250  for  1  yr.  9  mo.  15  da.  'dtl%, 

2.  $700  for  2  yr.  8  mo.  21  da.  at  5%. 

3.  1684.50  for  7  mo.  25  da.  at  6  %. 

4.  $1200  for  3  yr.  4  mo.  14  da.  at  5|  %. 

5.  $300  for  1  yr.  10  da.  at  8  %. 

6.  $45.75  for  2  yr.  8  mo.  12  da.  at  4  %. 

7.  $2500  for  9  mo.  13  da.  at  8%. 

8.  $560  for  2  yr.  3  mo.  23  da.  at  7  %. 

9.  $645.40  for  2  yr.  7  mo.  26  da.  at  9%. 

10.  $820.15  for  1  yr.  5  mo.  20  da.  at  4%. 

11.  $125  for  11  mo.  17  da.  at  8%. 

12.  $214.45  for  4  yr.  2  mo.  19  da.  at  6  %. 

13.  $750  for  1  yr.  6  mo.  28  da.  at  7  %. 

Find  the  interest  on  each  of  the  following  at  6%;  at 
6%;  at  7%;  at  4|%: 

14.  $800  from  Oct.  1,  1904  to  May  10,  1906. 

15.  $475  from  June  11,  1903  to  Nov.  18,  1904. 

16.  $240.60  from  April  8,  1904  to  Feb.  21,  1905. 

17.  $350  from  Jan.  1,  1904  to  Nov.  20, 1904. 

18.  $1340  from  June  8,  1903  to  Dec.  29,  1904. 

19.  $26.48  from  Sept.  12,  1905  to  Aug.  10,  1906. 

20.  $1700  from  March  24,  1905  to  Aug.  15,  1906. 

21.  $48.62  from  Nov.  18,  1902  to  July  20,  1904. 

22.  $5000  from  Sept.  7,  1903  to  Dec.  23,  1903. 

23.  $467.89  from  April  4,  1904  to  July  26,  1905. 


212  PERCENTAGE 

259.   Sixty  Days  Method. 

-1.  Money  loaned  for  less  than  one  year  is  usually 
loaned  for  90  da.,  60  da.,  or  less.  The  best  unit  of 
time  to  use  in  finding  the  interest  is  60  da.,  and  the 
best  rate  is  6%,  as  the  interest  at  6  %  for  60  da.  is  1  % 
(.01)  of  the  principal,  found  by  moving  the  decimal 
point. 

2.  Find  the  interest  on  $2700  for  60  da.  at  7%. 
Model  :     $  27       =  interest  at  6  %  for  60  da. 

4.50  =  interest  at  1  %  for  60  da. 
$  31.50  =  interest  at  7  %  for  60  da. 

3.  What  part  of  60  da.  is  30  da.?  10  da.?  20  da.? 
5  da.  ?  15  da.  ?  12  da.  ?  6  da.  ?  3  da.  ?  2  da.  ? 

4.  From  the  interest  for  60  da.,  how  may  the  interest 
be  found  for  30  da.  ?  for  15  da.  ?  for  6  da.  ?  for  20  da.  ? 
for  10  da.?  for  5  da.  ?  for  90  da.  ?  for  120  da.  ? 

5.  Find  the  interest  on  14000  at  6  %  for  60  da. ;  for 
90  da.  ;  for  30  da. ;  for  20  da. ;  for  120  da. 

6.  From  the  interest  for  30  da.,  how  may  the  interest 
be  found  for  15  da.?  for  45  da.  ?  for  10  da.  ?  for  5  da.  ? 

7.  From  the  interest  at  6%,  how  may  tlie  interest  be 
found  at  7  %?  at8%?  at  9  %  ?  at5%?  at  5J  %  ? 

8.  Find  the  interest  on  $500  for  90  da.  at  7%. 

Model  :  15=  interest  at  6  %  for  60  da. 

2.50  =  interest  at  6  %  for  30  da. 

$  7.50  =  interest  at  6  %  for  90  da. 

1.25  =  interest  at  1  %  for  90  da. 

9  8.75  =  interest  at  7  %  for  90  da. 

Find  the  interest  on : 
9.    $600    at  6  %  for  30  da.     11.    $1200    at  5  %  for  60  da. 
10.    $1000  at  7  %  for  90  da.     12.    $10,000  at  7%  for  45  da. 


INTEREST  213 

260.  Cancellation  Method.      (For  problems,  see  Sec.  258.) 

1.  What  part  of  360  da.  are  90  da.  ?  The  interest  for 
90  da.  is  what  part  of  the  interest  for  1  yr.  (360  da.)? 

2.  Find  the  interest  on  $600  for  90  da.  at  7  %. 

$150  ^0 

Model  :    $  ^^^  x  .07  x  f^  =  $  10.50. 

i 
The  interest  for  1  yr.  is  $  600  x  .07,  and  the  interest  for  90  da.  is 
/^,  or  ^  of  1600  X  .07. 

3.  What  part  of  1  year's  interest  is  the  interest  for 
6  mo.  ?  for  2  mo.  ?  for  8  mo.  ?  for  9  mo.  ?  for  10  mo.  ?  for 
15  da.  ?  for  45  da.  ?  for  1  mo.  15  da.  ?  for  3  mo.  20  da.  ? 
for  1  yr.  3  mo.  ?  for  1  yr.  6  mo.  ?  Find  the  interest  on 
$600  at  5%  for  each  of  these  periods. 

261.  Six  Per  Cent  Method.      (For  problems,  see  Sec.  258.) 

1.  Interest  is  sometimes  calculated  by  a  method 
commonly  known  as  the  Six  Per  Cent  Method.  By  this 
method,  the  interest  at  6  %  for  the  given  time  is  found, 
*and  from  this  the  interest  at  the  required  per  cent. 

2.  As  1  mo.  is  -^^  of  1  yr.,  the  rate  of  interest  for  1  mo. 
is  y^  of  6  %,  or  I  %  (,005);  and  as  1  da.  is  -^-^  of  1  mo.,  the 
rate  of  interest  for  1  da.  is  -^^  of  .005,  or  .000|^. 

The  interest  at  6%  for  1  yr.    =  .06  of  the  principal. 
The  interest  at  6%  for  1  mo.  =  .005  of  the  principal. 
The  interest  at  6%  for  1  da.   =  .000^  of  the  principal. 

3.  Find  interest  on  $350  for  2  yr.  7  mo.  21  da.  at  7 %. 

Model  :     Rate  for  2  yr.  at  6  % 12 

Rate  for  7  mo.  at  6  % 035 

Rate  for  21  da.  at  6  % .0035 

Rate  for  2  yr.  7  mo.  21  da 1585 

$350  X  .1585  =  $  55.475  =  int.  on  $350  for  2  yr.  7  mo.  21  da.  at  6  %. 

9.245+=  int.  on  $350  for  2  yr.  7  mo.  21  da.  at  1  %. 

$64.72      =  int.  on  $350  for  2  yr.  7  mo.  21  da.  at  7%. 


214  PERCENTAGE 

263.  Promissory  Notes. 

1.  A  written  promise  to  pay  a  definite  sum  of  money  at 
a  specified  time  is  called  a  promissory  note. 

A  promissory  note  is  usually  called  a  note. 

2.  Compare  the  promissory  note  on  p.  207  with  the  fol- 
lowing. The  note  on  p.  207  is  a  time  note,  as  the  time 
of  payment  is  specified  in  it. 

A  Demand  Note 


/ci'^^  €aUoinci,  ^aC,  fam,.  ^,  1907. 

€.^vci&yyuhncl,  for  value  received  <J  promise  to  pay  to 
fo^&^iA  R.  W^kite^,  or  order,  tkv&&  k^L'ylclA^&ci  cLo-ttoAj^, 
mith  interest  thereon  ai)  ^%  per   annum  from   date 

until  paid. 

^kcL'i.t&Q^  //.  5k(yYyux^. 


A  Joint  Note 


$600  ^taltU,  y>a^k.,  futif  /6,  1906. 

^ioctAf  dai^  after  date,  '^^,  o-v  &ttk&^  o-^  loQy,  promise 
to  pay  to  S^uin^  S^.  /CenA^cyyv,  or  order,  Q^Coc  kicr^iecC 
dattaiO',  with  interest  thereon  at  ^%  per  annum  from 
date  until  paid. 

Value  received.  W^cult&v  f.  BityvkeA,, 

fo-kn,  R.  Jbav^^e^. 


3.  Each  maker  of  a  joint  note  is  liable  for  its  payment 
in  full. 

4.  The  following  should  appear  in  a  note  : 

a.   The  time  and  place  where  the  note  was  executed. 
This  is  usually  written  at  the  top  and  toward  the  right 


PROMISSORY  NOTES  215 

h.  The  sum  to  be  paid,  including  the  rate  of  interest,  if 
any  is  paid. 

The  face  of  the  note  is  usually  written  in  figures  at  the  top  and 
toward  the  left  and  in  words  in  the  body  of  the  note. 

c.  The  signature  of  the  maker  or  makers. 

d.  The  time  of  payment. 

When  no  time  of  payment  is  specified,  the  note  is  payable  on 
demand. 

e.  Notes  usually  contain  the  words  value  received. 

Answer  each  concerning  the  two  notes  in  Sec.  262. 

5.  When  was  the  note  executed  ?     Where  ? 

6.  When  is  the  note  payable  ? 

7.  What  is  its  face  ? 

8.  Who  is  the  maker  ?  the  payee  ? 

9.  Is  the  note  negotiable  ? 

10.  When  a  note  becomes  due  it  is  said  to  mature.  In 
some  states  a  note  matures  three  days  after  the  time 
specified  in  the  note.  The  three  additional  days  are 
called  days  of  grace.  Days  of  grace  have  been  abolished 
in  most  states,  and  are  not  computed  in  the  answers  given 
in  this  book. 

11.  Each  state  has  fixed  its  own  legal  rate  of  interest, 
which  is  the  rate  allowed  on  claims  drawing  interest  when 
no  rate  of  interest  has  been  otherwise  arranged.  What 
is  the  legal  rate  in  the  state  in  which  you  live  ? 

12.  Many  states  have  fixed  a  maximum  rate  of  interest 
that  can  be  collected  by  agreement.  A  higher  rate  than 
that  authorized  by  law  is  called  usury.  Is  there  a  law 
against  usury  in  the  state  in  which  you  live  ? 

13.  Write  a  demand  note  ;  a  time  note  ;  a  joint  note. 


216  PERCENTAGE 

263.  Partial  Payments.  —  Mercantile  Rule.*  i.  On 
Jan.  1,  1906,  James  Smith  of  Los  Angeles,  Cal.,  borrowed 
of  Frank  Adams  $  1000  for  one  year  at  6  %,  giving  his  note 
for  this  amount.     Write  the  note. 

When  the  loan  was  made,  it  was  agreed  that  if  James  Smith  made 
any  payments  on  the  note  before  its  maturity,  he  would  be  credited 
with  each  partial  payment  and  would  be  credited  with  the  same  rate 
of  interest  as  he  was  paying,  6  %,  from  the  date  of  each  payment  until 
the  time  of  final  settlement. 

2.  On  July  1,  1906,  James  Smith  paid  Frank  Adams 
$400.  Indorse  this  payment  by  writing  "  July  1,  1906, 
f  400  "  on  the  back  of  the  note.  Final  settlement  was 
made  Jan.  1,  1907. 

Under  these  conditions  James  Smith  had  the  use  of  $1000  bor- 
rowed of  Frank  Adams  for  1  yr.,  and  Frank  Adams  had  the  use  ox 
$400  paid  by  James  Smith  for  6  mo.  (from  July  1,  1906,  to 
Jan.  1,  1907). 

At  the  time  of  final  settlement  James  Smith  owed  Frank  Adams 
$  1000,  with  interest  for  1  yr.,  at  6%,  or  $  1060  ;  and  Frank  Adams  owed 
James  Smith  $  400,  with  interest  for  6  mo.  at  6  %,  or  $  412.  In  settling 
the  note,  James  Smith  paid  Frank  Adams  $1060  -  $412,  or  $648. 

Notes  and  accounts  which  do  not  run  for  more  than  one  year,  on 
which  partial  payments  are  made,  are  often  settled  by  business  men 
as  above. 

Mercantile  Rule.  1.  Find  the  amount  of  the  faee 
of  the  note  at  the  time  of  settlement, 

2.  Find  the  amount  of  each  payment  from  the  date  of 
payment  to  the  date  of  settlement, 

3.  Subtract  the  sum  of  the  amounts  of  the  payments  from 
the  amount  of  the  face  of  the  note, 

3.  Write  a  note,  naming  some  pupil  as  payee  and  your- 
self as  payer.  Make  three  partial  payments  and  have 
them  indorsed  to  your  credit.     Settle  the  note. 

•  For  the  United  States  Rule  of  Partial  Payments,  see  Appendix,  p.  266. 


COMPOUND  INTEREST  217 

264.  Compound  Interest.* 

1.  When  the  unpaid  interest  is  added  to  the  principal, 
as  it  becomes  due,  to  form  a  new  principal  on  which  interest 
is  computed,  the  interest  is  called  compound  interest. 

Interest  may  be  added  to  the  principal  annually,  semiannually, 
quarterly,  etc.,  according  to  agreement. 

The  payment  of  compound  interest  cannot  usually  be  enforced  by 
law,  but  if  the  debtor  is  willing  to  pay  compound  interest,  it  may  be 
collected  without  violating  the  law  against  usury. 

2.  Savings  banks  generally  pay  interest  semiannually. 
When  it  is  not  collected  by  the  depositor,  it  is  added  to 
his  deposit  and  he  is  paid  compound  interest. 

3.  If  interest  is  collected  when  due  and  reinvested  at 
once  at  the  same  rate  of  interest,  the  result  is  the  same  as 
when  compound  interest  is  received. 

4.  Find  the  amount  of  |600  for  2  yr.  6  mo.  at  8%, 
interest  compounded  annually.  Find  the  difference  be- 
tween the  compound  interest  and  the  simple  interest. 

Model  :    f  600      =  principal  for  first  year. 

48       =  interest  for  first  year. 

$648      =  amount,  or  principal,  for  second  year. 

51.84  —  interest  for  second  year. 
$699.84  =  amount,  or  principal,  for  third  year. 

27.99  =  interest  for  6  mo. 
$727.83  =  amount  for  2  yr.  6  mo.  at  8%. 

Compound  interest  =  $727.83  -  $600  =  $127.83. 
Siniple  interest        =  120. 

Difference  =  $     7.83. 

5.  If  a  man  invests  $1000  at  compound  interest  at  6% 
when  he  is  30  years  of  age  and  keeps  it  earning  at  the 
same  rate  until  he  is  50  years  of  age,  what  will  be  the 
amount  of  the  f  1000  at  that  time  ?     (Use  table,  p.  320.) 

*  For  table  of  compound  interest,  see  Appendix,  p.  320. 


218  PERCENTAGE 

265.  Bank  Discount  and  Proceeds. 

1.  Banks  usually  collect  interest  in  advance  on  sums 
loaned.  Thus,  if  George  White  borrows  $100  at  a  bank 
for  60  da.  at  6%,  his  note  will  be  made  out  for  flOO, 
and  the  bank  will  deduct  from  this  amount  the  in- 
terest on  $100  for  60  da.  at  6%,  or  $1.  Mr.  White  wiU 
receive  $99.  At  the  end  of  60  da.  he  will  pay  the  bank 
the  face  of  the  note,  or  $100. 

2.  If  interest  is  collected  in  advance,  how  much  money 
will  a  person  receive  at  a  bank  on  a  note  for  $2000  for  60 
da.,  if  the  bank  charges  8  %  interest  ? 

3.  On  April  8  J.  J.  Dow  bought  $  600  worth  of  goods 
of  D.  C.  Brown,  on  90  da.  time,  giving  his  note  for  the 
amount  without  interest.  On  the  same  day  D.  C.  Brown 
sold  the  note  to  a  bank,  the  bank  deducting  6  %  interest 
for  the  term  of  the  note  (90  da.).  Find  the  amount 
received  for  the  note  by  D.  C.  Brown. 

4.  Interest  paid  in  advance  upon  the  amount  due  on  a 
note  at  its  maturity  is  called  bank  discount.  Bank  dis- 
count is  computed  from  the  date  of  the  purchase  of  the 
note  by  the  bank  to  the  legal  date  of  maturity. 

Some  banks  include  both  the  day  of  purchase  and  the  day  of 
maturity  in  the  discount  period.  "When  days  of  grace  are  allowed, 
these  are  included  in  the  discount  period. 

5.  The  sum  paid  for  a  note  when  sold  is  called  the 
proceeds  of  the  note.  The  proceeds  on  a  note  is  the 
amount  due  at  maturity,  less  the  bank  discount. 

6.  C.  W.  Smith  held  a  note  against  R.  E.  Orr  for 
$4000  for  60  da.  without  interest.  After  20  da.,  he  sold  it 
to  a  bank  at  a  discount  of  6  %  ;  that  is,  the  bank  deducted 
6  %  int.  on  the  note  for  the  40  da.  between  its  purchase  and 
expiration.     Find  the  bank  discount  and  the  proceeds. 


BANK  DISCOUNT  219 

7.  On  April  24,  1906,  James  J.  Hall  sold  a  horse  to 
G.  M.  Bruce  for  f  150,  taking  in  payment  his  note  for  1 
year  with  interest  at  6  % .  Find  the  amount  of  the  note 
at  maturity. 

8.  Mr.  Hall  (Ex.  7)  needed  money,  so  he  sold  the  note 
to  a  bank  on  the  same  day,  the  bank  discounting  it  at 
6  % .     How  much  did  Mr.  Hall  receive  for  the  note  ? 

Model  :  Face  of  note  =  $  150 

Interest  for  1  yr.  at  6  %  =  9 

Amount  at  maturity       =  $  159. 

Discount  1  yr.  at  6  %       =        9.54  (computed  on  $  159) 

Proceeds  =  $  149.46 

9.  If  the  note  (Probs.  T  and  8)  had  been  discounted 
at  8  %  instead  of  6  % ,  what  amount  would  Mr.  Hall  have 
received  ? 

10.  If  the  note  (Probs.  7  and  8)  had  been  discounted 
three  months  after  date  of  issue,  or  on  July  24,  1906,  the 
bank  would  have  deducted  interest  on  the  amount  due  at 
maturity  (1 159)  for  the  exact  number  of  days  from  July 
24,  1906  to  April  24,  1907  (7  da.  4-  31  da.  +  30  da.  +  31 
da.  4-  30  da.  +  31  da.  +  31  da.  +  28  da.  +  31  da.  +  24 
da.),  or  for  274  da.  Find  the  amount  which  Mr.  Hall 
would  have  received. 

11.  A  90-da.  note  for  $  500,  without  grace,  dated  Aug. 
5,  1905,  with  interest  at  5%,  was  discounted  at  a  bank  on 
Aug.  25  at  6%.  Find  the  day  of  maturity,  the  amount 
at  maturity,  the  bank  discount,  and  the  proceeds. 

12.  A  man  borrowed  flOOO  of  a  bank  for  1  yr.  at  6%, 
paying  interest  in  advance.  6%  interest  in  advance  on 
$1000  is  equivalent  to  what  rate  paid  at  the  end  of  the 
year  ? 


MO  PERCENTAGE 

266.  Review. 

1.  During  a  certain  school  month  a  boy  worked  209 
problems,  of  which  194  were  correct.  Find  the  per  cent 
of  correct  work. 

2.  A  baseball  team  won  43  games  and  lost  15  games 
one  season.     Find  the  per  cent  of  games  won. 

3.  The  rent  of  a  house  was  raised  from  $  30  a  month 
to  I  35 ;  this  was  an  increase  of  what  per  cent  ? 

4.  A  person  bought  a  house  for  $6000.  The  taxes, 
insurance,  repairs,  and  other  expenses  connected  with  the 
property  amounted  to  $120  a  year.  For  how  much  a 
month  must  the  property  be  rented  to  net  6%  on  the 
investment  ? 

5.  A  man  built  two  flats  costing  him  $  5000  on  a  lot 
which  cost  $2000.  He  rented  one  of  the  flats  for  $40  a 
month  and  the  other  for  $35.  The  expenses  connected 
with  the  property  amounted  to  $200  a  year.  The  net 
income  amounted  to  what  per  cent  on  the  investment  ? 

6.  An  electric  light  meter  registered  80,000  watt 
hours  on  Oct.  9,  and  106,000  watt  hours  on  Nov.  9.  Find 
the  amount  of  the  bill  for  the  month  at  9^  for  each  1000 
watt  hours. 

7.  A  gas  meter  registered  29,800  cu.  ft.  on  April  24, 
and  31,800  cu.  ft.  oh  May  24.  Find  the  amount  of  the 
bill  for  the  month  at  $  .90  per  1000  cu.  ft. 

8.  The  population  of  a  certain  city  was  47,235  in  1900, 
and  60,624  in  1910.     Find  the  increase  per  cent. 

9.  At  66 ff  a  sack  (100  lb.),  what  is  the  price  of  coal 
per  ton  ? 

10.    Find  the  interest  on  $2800  from  June  8  to  Jan.  16 
at  6  %  per  annum. 


PART   IV 

FORMS   AND  MEASUREMENTS* 

267.     1.    Lines  are  vertical      j  ,   horizontal    — ,    and 
oblique    \  /. 

2.  These  are  right  angles.  | |  [  | 

3.  A  rectangle  has  four  right  angles.  QJ  |        [ 

4.  These  are  right  triangles.  \yx\ 

5.  These  are  acute  angles.    /\    v 

6.  These  are  acute-angled  triangles.  ^  \J 

7.  These  are  obtuse  angles.  ^-^"-\     ^^..^ 

8.  These  are  obtuse-angled  triangles.    ^-^^      ^^^Z^ 

9.  Perpendicular    (p)    means   at   right         .     |^^ 
angles  to.  -1^  l^ 

10.    These  figures  have  a  base   (5)  ^i        i^     y^a     A 
and  an  altitude  (a).  ^-t         "^""^       b" 


11.    These  lines  are  parallel. 


12.  These  are  quadrilaterals.       V~\  /~~J  /     \  ^    | 

13.  These  quadrilaterals  are  parallelograms.    I     I    I      7 

13  ^Pl^] 


14.    These  are    rectangular 
prisms. 

♦  With  complete  reviews. 
221 


222 


FORMS  AND  MEASUREMENTS 


15.    These  are  triangular 
prisms. 


16.    Circumference,  diameter,  and  radius  belong 
to  the  circle. 


17.    These  are  cylinders. 


B  ^ 


circle. 


268.  Relation  of  Forms. 

Study  the  relation  of  these  forms 


L 


Right  Angle. 


Rectanqle.  Rectangular  Prism. 


L      [X 


Right  Angle.  Right  Tbl^u^gle.  Right  Triangular  Prism. 


A        A 


Acute  Angle.      Acute  Triangle.     Acute  Triangular  Prism. 


Obtuse  Angle.     Obtuse  Triangle.   Obtuse  Triangular  Prism. 


O 


Cl&CLiB. 


m 

Sphere. 


Ctlindbb. 


LINES  AND   ANGLES  223 

269.  Lines. 

1.  Lines  that  extend  in  the  same  direction  and  are  the 

same  distance  apart  are  called  parallel  

lines.  — 

2.  Suspend    a    weight  by   a   string.        p^^allel  Lines. 
When  the  weight  is  at  rest,  the  line  represented  by  the 
string  is  called  a  vertical  line. 

3.  The  surface  of  the  water  in  a  tank  or  a  pond  is  said 
to  be  level,  or  horizontal.  A  slanting  line  is  called  an 
oblique  line. 

A  vertical  line  is  represented  on  a  page  by  a  line  parallel  to  the 
sides,  and  a  horizontal  line  by  a  line  parallel  to  the  top  and  bottom. 

4.  Hold  your  pencil  in  a  vertical  position  ;  in  a  hori- 
zontal position  ;  in  an  oblique  position, 

5.  Point  to  surfaces  in  the  schoolroom  that  are  hori- 
zontal, vertical,  oblique,  parallel. 

6.  Draw  two  vertical  parallel  lines  on  the  blackboard; 
two  horizontal  parallel  lines;  two  oblique  parallel  lines. 

270.  Right  Angles. 

1.  Two  lines  that  meet  form  an  angle,  Z.  When  two 
lines  form  a  square  corner,  the  angle  between  them  is 
called  a  right  angle. 

2.  Draw  four  right  angles.  .  .  . —  — . 

3.  Point  to  surfaces  in  your  school-    — -  — '  '  ^ 
room  that  meet  at  right  angles. 

4.  Two  lines  that  form  a  right  angle  are  said  to  be  per- 
pendicular to  each   other.     Draw   perpen-       i     \_p_    p/ 

dicular  lines.  — ■—  '         ^ 

Perpendiculab 

5.  Point    to    lines    or    surfaces   in   the  Lines. 
schoolroom  that  are  perpendicular  to  each  other. 


224  FORMS  AND  MEASUREMENTS 


n 


n 


Square.  Ilurizontal.  Vertical. 

Bectanolbs. 


271.  Rectangles. 

1.  A  figure  whose  angles 
are  all  right  angles  is  called 
a  rectangle.  Reel  means 
right.  Rectangle  means 
having  right  angles. 

2.  A  rectangle  whose  sides  are  all  the  same  length  is 
called  a  square. 

A  rectangle  having  two  opposite  sides  longer  than  the  other  two 
opposite  sides  is  sometimes  called  an  oblong. 

3.  Draw  a  square ;  a  vertical  rectangle ;  a  horizontal 
rectangle.  Point  to  surfaces  in  your  schoolroom  that  are 
rectangles.     Are  any  of  these  squares? 

4.  How  many  sides  has  a  rectangle  ?  How  many  angles 
has  a  rectangle?  Are  the  sides  of  a  rectangle  parallel? 
Name  surfaces  not  in  your  schoolroom  that  are  rectangles. 

272.  1.  Draw  a  square  whose  side  is  1  foot.  This  is 
called  a  square  foot.  Draw  and  name  a  square  whose  side 
is  1  inch. 

2.  Draw  a  square  yard.  Divide  it  into  square  feet. 
How  many  square  feet  are  there  in  a  square  yard? 

3.  Divide  a  square  foot  into  square  inches.  How  many 
square  inches  are  there  in  a  square  foot  ? 

4.  A  square  16 J  feet  each  way  is  called  a  square  rod. 
Mark  out  a  square  rod  on  the  school  grounds. 

5.  Draw  a  square  whose  side  is  2  inches.  Divide  it 
into  square  inches.     How  many  are  there  ? 

6.  Draw  3  inch  squares  and  a  3-inch  square.  Compare 
the  size  of  a  3-inch  square  and  3  square  inches. 

7.  The  number  of  square  units  in  a  surface  is  called 
its  area. 


RECTANGLES  225 

273.  Areas  of  Rectangles. 

1.  Repeat  the  Table  of  Linear  Measure  (§  99).  Re- 
view p.  81.     Repeat  the  Table  of  Square  Measure  (§  101). 

2.  Using  the  scale  J  in.  =  1  rd.,  make  a  drawing  to 
represent  a  rectangle  16  rd.  long  and  10  rd.  wide.  Ex- 
press the  area  of  the  rectangle  in  acres. 

3.  Find  the  area  of  a  flower  bed  that  is  12  ft.  9  in. 
long  and  8  ft.  4  in.  wide. 

4.  How  many  acres  are  there  in  a  tract  of  land  80  rd. 
by  80  rd.  ? 

5.  A  farm  that  contains  80  A.  is  J  mi.  wide.  How 
long  is  the  farm  ? 

6.  Find  the  number  of  square  yards  of  surface  in  the 
walls  and  ceiling  of  your  schoolroom,  deducting  for  the 
doors  and  windows. 

7.  Find  the  value  of  a  field  40  rd.  long  and  20  rd.  wide 
at  f  85  an  acre. 

274.  Written  Exercises. 

1.  Reduce  2  yd.  2  ft.  7  in.  to  inches. 

2.  Find  the  sum  of  8  ft.  6  in.,  7  ft.  4  in.,  9  ft.  11  in., 
and  6  ft.  5  in. 

3.  Find  the  perimeter  of  a  rectangle  whose  length  is 
24  ft.  8  in.  and  whose  width  is  15  ft.  10  in. 

4.  How  many  rods  of  fence  are  required  to  inclose  a 
rectangular  20-acre  field  whose  length  is  80  rd.  ? 

5.  How  many  bundles  of  shingles  are  necessary  to 
shingle  a  surface  50  ft.  by  16  ft.,  if  the  shingles  are  laid 
4  in.  to  the  weather  ? 

6.  How  many  yards  of  carpet  are  necessary  to  cover  a 
floor  16  ft.  by  12  ft.,  if  the  carpet  is  27  in.  in  width  ? 

AB.  — 15 


226  FORMS  AND  MEASUREMENTS 

275.  Right  Triangles. 

1.  Draw  a  right  angle.     The  point  at  which  the  lines 
meet  is  called   the  vertex  of    the  angle. 

2.  Draw  a  rectangle.  Draw  a  straight  line  joining  the 
vertices  of  the  opposite  angles  of  the  rec- 
tangle. This  line  is  called  the  diagonal  of 
the  rectangle.  The  diagonal  divides  the 
rectangle  into  two  equal  triangles. 

3.  A  figure   having   three  angles  is  called  a  triangle. 
Tri  means  three.     Triangle  means  having  three  angles. 

4.  A  triangle  having  one  right  angle  is  called  a  right 
triangle. 

276.  Area  of  Right  Triangles. 

1.    The  base  of  a  figure  is  the  side  on  which  it  is  as- 
6  sumed  to  rest,  and  the  altitude  is  the 

perpendicular  distance  between  the  top 
a  and  the  base,  or  the  base  produced. 

Consider  the  length  of  a  rectangle  as  its 
and  the  width  as  its  altitude. 


6 

RiaHT  Triangles. 


2.  The  area  of  a  right  triangle  is  what  part  of  the  area 
of  a  rectangle  having  the  same  base  and  altitude? 

The  area  of  a  right  triangle  is  equal  to  one  half  the  prod- 
uct of  its  base  and  altitude. 

The  work  is  sometimes  shorter  if  the  altitude  is  multiplied  by  one 
half  the  base,  or  if  the  base  is  multiplied  by  one  half  the  altitude. 

Dimensions  must  be  expressed  in  like  units.  By  the  product  of 
the  lines  is  meant  the  product  of  the  numbers  denoting  them. 

3.  Find  the  area  of  right  triangles  of  the  following 
dimensions:  base,  12  in.,  altitude,  8  in. ;  base,  6J  ft.,  alti- 
tude, 9 J  ft.  ;   base,  40  rd.,  altitude,  80  rd. 


Parallelogram. 


PARALLELOGRAMS  227 

277.  Parallelograms. 

1.  A  plane  (flat)  figure  bounded  by  four  straight  lines 
is  called  a  quadrilateral.  Quadri  means  four  and  lateral 
means  sides.     Quadrilateral  means  having  four  sides. 

2.  A  quadrilateral  whose  opposite  sides  are  parallel  is 
called  a  parallelogram.  Is  a  rectangle  a  parallelogram? 
Draw  a  quadrilateral  that  is  not  a  parallelogram. 

3.  This  figure  represents  a  city  lot.  Is  the  form  of  the 
lot  a  parallelogram  ?  The 

form  of  the  lot  may  be 
regarded  as  composed  of 
a  rectangle,  a^  and  two 
right  triangles,  b  and  o. 
Since  the  figure  is  a  par- 
allelogram, triangle  h  is  the  same  size  as  triangle  c.  If 
triangle  h  were  cut  off  and  placed  alongside  triangle  c, 
with  the  side  fd  on  the  side  ge^  what  change,  if  any, 
would  this  make  in  the  size  of  the  lot  ?  What  change,  if 
any,  would  it  make  in  the  form  of  the  lot?  What  would 
be  the  dimensions  of  the  resulting  lot?  How  should  you 
find  its  area? 

4.  The  sum  of  the  areas  of  triangles  h  and  c  is  equiva- 
lent to  the  area  of  a  rectangle  having  the  same  base  and 
altitude  as  the  triangles.  Therefore  the  number  of  square 
feet  in  the  lot  may  be  found  by  multiplying  120  by  40. 

The  area  of  a  parallelogram  is  equal  to  the  product  of  its 
base  and  altitude. 

5.  Cut  several  parallelograms  out  of  paper,  and  show  by 
the  method  of  Prob.  3  that  each  parallelogram  has  the 
same  area  as  a  rectangle  having  the  same  base  and  altitude. 

6.  Draw  several  parallelograms.  Assign  the  dimen- 
sions, and  find  the  area  of  each. 


"5- 

"     K 

26rd. 
Fig.  1.— Tkapkzoid. 

228  FORMS  AND  MEASUREMENTS 

278.   Trapezoids. 

1.  Fig.  1  represents  a  field  whose  area  is  to  be  found.  It 
may  be  regarded  as  composed  of  a  rectangle,  a,  and  a  right 

2Q    ,  triangle,  h.     Its  area  is  the  sum 

of  the  areas  of  these  two  parts. 
The  number  of  square  rods  in  the 
rectangle  is  10  x  20,  and  in  the  tri- 
angle is  10  X  3.  The  number  in 
the  entire  field  is  10  x  23.  Explain. 

2.  What  is  the  average  length  of  the  two  sides  of  the 
field  (Fig.  1)  ?  The  field  has  the  same  area  as  a  field  of 
the  same  width  whose  length  is  one  half  of  the  sum  of 
20  rd.  and  26  rd.,  or  23  rd.     Explain. 

3.  Draw  five  figures  similar  to  Fig.  1.  Assign  the 
dimensions,  and  find  the  area  represented  by  each. 

4.  Figure  2  represents  a  cross  section  of  a  foundation 
wall.  It  may  be  regarded  as  composed  of  a  rectangle,  a, 
and  two  right  triangles,  h  and  c.  What  is  the  combined 
length  of  the  bases  of  the  two  triangles  ?  If  triangle  h 
were  cut  off  and  placed  in  an  inverted  position  alongside 
triangle  c,  what  change  would  it  make  in  the  form  of  the 
figure  ?  What  would  be  the  dimensions  of  the  resulting 
figure  ?     How  would  you  find  its  area  ? 

5.  The  area  of  triangle  h  (Fig.  2)  is  equivalent  to  the 
d      4ft      e       area  of  a  rectangle  of  the  same  altitude 

whose  base  is  one  half  of  the  base  of 
the  triangle.  Is  the  same  true  of  the 
area  of  triangle  c  2  The  area  of  the  sur- 
face represented  by  the  entire  figure  is 
equivalent  to  the  area  of  a  rectangle  of 
Fio.  2.  the  same  altitude  but  whose  base  is  one 

half  the  sum  of  the  two  bases  de  and  fg. 


TRIANGLES 


6.  A  quadrilateral  that  has  only  two  sides  parallel  is 
called  a  trapezoid.     See  Figs.  1  and  2,  p.  228. 

The  area  of  a  trapezoid  is  equal  to  the  product  of  its  alti- 
tude and  one  half  the  sum  of  its  bases. 

7.  Draw  five  figures  similar  to  Fig.  2,  p.  228.  Assign 
the  dimensions,  and  find  the  area  of  each. 

The  area  of  any  quadrilateral  may  be  found  by  resolving  it  into 
triangles  or  into  rectangles  and  triangles. 

279.  Area  of  Triangles. 

1.  An  angle  that  is  less  than  a  right 
angle  is  called  an  acute  angle.  Acute 
means  sharp  or  pointed. 

2.  An  angle  that  is  greater  than  a 
right  angle  is  called  an  obtuse  angle. 
Obtuse  means  dull  or  blunt. 

3.  A  triangle  whose  angles  are 
all  acute  is  called  an  acute-angled 
triangle. 

4.  A  triangle  one  of  whose  an- 
gles is  obtuse  is  called  an  obtuse- 
angled  triangle.  Obtuse-anqled  triangles. 

5.  Can  you  draw  a  triangle  having  more  than  one  ob- 
tuse angle  ?  Draw  an  acute-angled  triangle;  an  obtuse- 
angled  triangle. 

6.  Any  triangle  may  be  divided  into  two  right  triangles. 


Acute  Angles. 


Obtuse  Angles. 


AV 

Acute-angled  TblAuNGles. 


7.    Triangle  Imn  in  each  of  the  above  figures  is  one  half 
of  the  rectangle  opln.     Explain. 


^-80 


FORMS  AND  MEASUREMENTS 


8.    Any       triangle 
may  be  considered  as 
one  half  of  a  parallelo- 
a  c  gram  of  the  same  base 

Fig-  1-  and  altitude.  Triangle 

ahe  in  Fig.  1  is  one  half  of  the  parallelogram  ahdc. 

9.  Draw  five  triangles.  Show  by  the  methods  given 
in  Probs.  T,  8  that  the  triangles  are  each  one  half  of  a 
parallelogram  having  the  same  base  and  altitude. 

The  area  of  a  triangle 
is  equal  to  one  half  the 
product  of  its  base  and 
altitude. 

10.  Draw  five  trian- 
gles. Assign  their  di- 
mensions and  find  the 
area  of  each. 

The  area  of  any  figure 
may  be  found  by  resolving 
it  into  triangles. 

11.  Figure  2  repre- 
sents a  farm  80  rd.  by 
160  rd.  Find  the  area 
of  each  field. 

12.  The  two  sides  of 
a  field,  one  132  rods, 
the  other  152  rods  in 
length,  are  parallel. 
The  perpendicular  dis- 
tance between  the 
two  sides   is   80   rods. 

FiQ.  2.  Diagram  and  find  area. 


r-'^ 

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40rd. 

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t 

eOrd. 

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E 

CU 

J 

40  rd. 

t 

/ 

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f 

05 

^ 

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t 
5 

25  rd. 

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^         D 

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V. 

/ 

^ 

/ 

15  rd 

/ 

I'/ 

40  rd. 

O) 

'^   / 

.      ^      ^ 

/ 

^' 

^ 

§               J 

B 

? 

40rd 

/           ^ 

^ 
^ 

/                          80  rd 

<\j 

CIRCLES  231 

280.  Circles.  . 

1.  Draw  a  circle.  Mark  the  center  ((y)  (^  "^ 
of  the  circle.  The  line  bounding  the  circle 
is  called  the  circumference.  Draw  a  line 
from  the  center  of  the  circle  to  the  cir- 
cumference. This  is  called  a  radius  of 
the  circle.  Draw  a  straight  line  through  the  center  of 
the  circle  from  circumference  to  circumference.  This  is 
called  the  diameter  of  the  circle.  Compare  the  length  of 
the  diameter  with  the  length  of  the  radius. 

2.  Measure  the  diameter  of  a  circle.  Measure  the  cir- 
cumference of  the  same  circle.  Divide  the  circumference 
by  the  diameter.  The  answer  should  be  nearly  3.1416 
(3|).  This  is  the  ratio  of  the  circumference  to  the 
diameter.  This  ratio  is  commonly  denoted  by  the  symbol 
TT,  which  is  a  Greek  letter  named  pi. 

3.  The  circumference  of  a  circle  is  3.1416  times  the  di- 
ameter. How  can  you  find  the  circumference  when  the 
radius  is  given  ?  When  the  circumference  is  given,  how 
can  you  find  the  diameter  ?  the  radius  ? 

4.  If  a  wagon  wheel  is  3|  ft.  in  diameter,  what  is  its 
circumference  ?  How  many  times  will  it  turn  in  going  1 
mile  ? 

5.  Find  the  circumference  of  a  circle  whose  diameter  is 
24  in. ;  7J  in.  ;  2  ft.  6  in. ;  40  mi.  ;  80  rd. 

6.  Find  the  circumference  of  a  circle  whose  radius  is 
2  in.  :    2  ft.  4  in.  ;  6  ft.  ;  40  rd.  ;  3J  yd. 

7.  Find  the  diameter  and  radius  of  a  circle  whose  cir- 
cumference is  24  ft.  ;  12 J  in.  ;  1  mi.  ;  36  rd. 

8.  Find  the  equatorial  diameter  of  the  earth  if  its 
equatorial  circumference  is  24,900  mi. 


232  FORMS  AND  MEASUREMENTS 

281.  Area  of  a  Circle. 

1.  A  circle  may  be  regarded  as  com- 
posed of  an  infiuite  number  of  triangles, 
the  sum  of  whose  bases  is  the  circumfer- 
ence of  the  circle  and  whose  altitude  is 
the  radius  of  the  circle.  Therefore  the 
area  of  a  circle  is  the  area  of  the  triangles 
composing  it. 

The  area  of  a  circle  is  equal  to  one  half  the  product  of  its 
circumference  and  radius. 

2.  The  circumference  of  a  circle  is  3.1416  times  the 
diameter,  or  tt  times  2  times  the  radius  =  2  7rr. 

Area  of  circle  (Prob.  1)  =  ^lli2ij:. 

Substituting  2  irr  for  Cir., 

Area  of  circle  =  - — =  tt  x  r  x  r,  or  tt  r^,  read  pi  r  square. 

The  area  of  a  circle  is  equal  to  irr^  (3.1416  X  r  x  r). 

3.  Find  the  area  of  a  circle  whose  radius  is  2  in. ;  12 
in.  ;  24  ft.  ;  4.5  rd.  ;  64  yd.  ;  18  ft.  ;  3  ft.  7  in.;  8|  in. 

4.  Find  the  area  of  a  circle  whose  circumference  is  24 
in.  ;  2  ft.  4  in.  ;  40  rd.  ;  28  yd. ;  J  mi.  ;  1  mi. 

5.  The  diameter  of  a  circular  flower  bed  is  6  ft.  What 
is  its  area  ? 

6.  The  atmospheric  pressure  is  about  15  lb.  to  the 
square  inch.  Find  the  pressure  on  a  surface  of  a  watch 
crystal  \\  in.  in  diameter. 

7.  Over  how  many  square  feet  of  surface  can  a  horse 
graze  when  tied  with  a  rope  20  ft.  long  ? 

8.  Which  has  the  greater  surface,  a  rectangular  table 
top  that  is  3'  6"  by  3'  6''  or  a  circular  table  top  that  is  3' 
9"  in  diameter? 


PRISMS  AND  CYLINDERS 


233 


282.  Volume  of  Prisms  and  Cylinders. 

1.  Draw  a  rectangle.  A  rectangle  has  two  dimensions; 
namely,  length  and  width.  If  it  is  given  a  third  dimension, 
thickness,  it  becomes  a  rectangular  solid  or  a  rectangular 
prism. 

2.  Anything  that  has  length,  breadth,  and  thickness  is 
called  a  solid. 


/^ 

hi 

Rectangles. 


Rectangular  Prisms. 


3.  How  many  rectangular  faces  has  a  rectangular 
prism  ?   Name  some  rectangular  prisms  that  jo\i  have  seen. 

4.  A  rectangular  prism  whose  faces  are  all  equal  squares 
is  called  a  cube. 

5.  Construct  rectangular  prisms  out  of  cardboard  or 
paper. 

283.  1.  Draw  a  triangle.  How  many  dimensions  has  a 
triangle  ?  If  it  is  given  a  third  dimension,  it  becomes  a 
triangular  prism.  The  ends  of  a  triangular  prism  are  tri- 
angles and  the  sides  are  rectangles. 


Triangles. 


Triangular  Prisms. 


2.  How  many  faces  has  a  triangular  prism  ?    How  many 
of  the  faces  are  triangles  ?     How  many  are  rectangles  ? 

3.  Construct  a  triangular  prism   out  of  cardboard  or 
paper. 


234 


FORMS  AND  MEASUREMENTS 


284.  1.  Draw  a  circle.  If  a  circle  is  given  three  di- 
mensions, it  becomes  a  cylinder.  Mention  some  cylindrical 
objects. 


Cylinders. 


2.  Construct  a  cylinder  out  of  paper. 

3.  The  number  of  cubic  units  that  a  solid  contains  is 
called  its  volume,  or  capacity. 


s  ^ 


9  square  units. 
Fig.  1. 


9  cubic  units. 
Fig.  2. 


8  times  9  cubic  unlta. 
Fig.  3. 


285.  1.  The  area  of  the  end,  or  base,  of  a  prism  or  a 
cylinder  tells  how  many  square  units  that  surface  contains 
(Fig.  1).  There  are  as  many  cubic  units  in  one  unit  of 
length  as  there  are  square  units  in  the  surface  of  the  end, 
or  base  (Fig.  2).  Explain.  There  are  as  many  cubic 
units  in  the  prism  or  cylinder  as  the  product  of  the 
number  of  units  in  the  area  of  the  end,  or  base,  and  the 
number  of  units  in  the  length  or  altitude  of  the  prism  or 
cylinder  (Fig.  8).     Explain. 

2.  The  volume  of  a  prism  or  a  cylinder  is  equal  to  the 
product  of  the  area  of  the  end,  or  base,  and  the  length, 
or  altitude. 


PRISMS  AND  CYLINDERS  236 

3.  Find  the  capacity  of  a  cylindrical  tank  whose  diameter 
is  18  in.  and  whose  height  is  4  ft.  6  in. 

Number  of  square  feet  in  area  of  base  =  3.1 416  x  |  x  f(|  X  |  =  r'^). 
Number  of  cubic  feet  in  capacity  =  3.1416  x  |  x  |  x  4|. 

4.  There  are  231  cubic  inches  in  a  gallon.  How  many 
gallons  will  the  tank  (Prob.  3)  hold  ? 

5.  Find  the  number  of  cubic  feet  in  a  bin  8  feet  long, 
4  feet  wide,  and  6  feet  deep. 

6.  There  are  2150.42  cubic  inches  in  a  measured  bushel. 
How  many  bushels  will  the  bin  (Prob.  5)  hold  ? 

7.  Find  the  number  of  cubic  feet  of  air  in  a  room  16 
feet  long,  10  feet  wide,  and  9  feet  high. 

8.  Find  the  number  of  cubic  yards  of  earth  that  must 
be  removed  in  excavating  a  basement  8  feet  deep,  36  feet 
long,  and  24  feet  wide. 

286.    Surfaces  of  Prisms  and  Cylinders. 

1.  How  many  surfaces  has  a  cube?  a  rectangular 
prism  ?  a  triangular  prism  ?  Construct  each  out  of  card- 
board. State  how  the  area  of  the  combined  surfaces  of  a 
prism  may  be  found. 

2.  Find  the  area  of  the  surfaces  (excluding  the  ends)  of 
a  timber  12"  by  12"  and  16'  in  length.  What  name  is 
given  to  such  a  solid  ? 

3.  Bring  together  the  ends  of  a  sheet  of  paper  so  that 
the  sides  form  circles.  What  name  is  given  to  the  fig- 
ure formed  by  the  sheet?  The  length  of  the  sheet  be- 
comes the  circumference  of  the  base  of  the  cylinder  and 
the  width  of  the  sheet  becomes  its  altitude.  The  area  of 
the  cylinder  (excepting  the  bases)  is  therefore  the  area  of 
the  rectangle  forming  its  convex  surface. 

4.  Find  the  convex  surface  of  a  cylindrical  tank  whose 
diameter  is  6  feet  and  whose  altitude  is  8  feet. 


286 


FORMS  AND  MEASUREMENTS 


287.    Written  Problems. 

1.  If  it  takes  .98  cu.  yd.  of  crushed  stone,  .47  cu.  yd. 
of  sand,  and  1.56^hbl^of  cement  to  make  1  cu.  yd.  of 
concrete,  how  much  di  each  will  make  100  cu.  yd.  of 
concrete  ? 

\y  2.  How  much  crushed  stone  (Prob.  1),  sand,  and 
cement  will  be  required  to  build  a  concrete  wall  6  ft. 
high,  18  in.  thick,  and  60  ft.  long,  if  3j  cu.  ft.  of  cement 
is  a  barrel?  (Express  each  answer  as  a  whole  number, 
since  a_fractional  part  of  these  units  cannot  be  purchasedL)_^- 

3.  If  the  sand  costs  80^  per  cubic  yard,  the  cement  $2 
per  barrel  of  3|  cu.  ft.  each,  the  stone  $1.50  per  cubic 
yard,  and  the  labor  for  building  the  wall  80^  per  cubic 
yard,  find  the  cost  of  the  wall  (Prob.  2). 

4.  Mr.  Adams  owns  a  50-ft.  lot  on  which  he  has  re- 
cently built  a  house.  He  wishes  to  have  a  6-ft.  cement 
sidewalk  laid  along  the  front  of  the  lot  2  ft.  in  from  the 


D 


( 


^ 


curb,  and  a  4-ft.  cement  sidewalk  laid 
from  the  curb  to  the  front  steps  of  the 
house,  20  ft.  in  from  the  curb.  Find 
the  cost  of  the  walks  at  $.12 J  per  square 
foot.  First  make  a  diagram  of  the 
walks. 


95' 


73 


5.  The  figure  represents  a  lot  owned 
by  Mr.  Morse.  Find  the  area  of  the  lot. 
Find  the  cost  of  excavating  a  basement 
8  ft.  deep  on  the  property  at  $1.25  per  cubic  yard. 

6.  How  much  will  it  cost  at  $  .32  a  cubic  yard  to  re- 
move 1  ft.  of  dirt  from  a  lot  100  ft.  by  150  ft.  ? 

7.  At  $5.50   a   cord,  how  much  will  a  pile  of  wood 
24  ft.  long,  4  ft.  wide,  and  8  ft.  high  cost? 


CIRCULAR  MEASUREMENTS 


237 


MEASUREMENT  OF  CIRCLES  AND  CIRCUMFERENCES 

288.  1.  For  the  purpose  of  measurement  circumfer- 
ences of  circles  are  considered  to  be  divided  into  360 
equal  parts,  called  degrees  (°). 

2.  A  portion  of  a  circumference  is  called  an  arc. 

3.  What  portion  of  the  circumfer- 
ence of  the  larger  circle  is  arc  a5? 
What  portion  of"  the  circumference  of 
the  smaller  circle  is  arc  dV  ? 

4.  How  many  degrees  are  there  in 
arc  ah  ?  in  arc  ac  ?  in  arc  aJW  ?  in 
arc  a! c^  ? 

5.  If  the  circumference  of  the  larger  circle  is  24,90^0. 
mi.,  how  long  is  each  degree  on  the  circumference  ?  If 
the  circumference  of  the  smaller  circle  is  6000  mi.,  how 
long  is  each  degree  on  the  circumference  ? 

6.  As  the  angle  at  0^  the  common  center  of  the  two 
circles,  increases  or  diminishes  as  fast  as  the  arc  suspended 
by  its  sides  increases  or  diminishes,  the  angle  is  also 
measured  in  degrees.  Thus,  when  the  arc  between  two 
radii  is  90°,  the  angle  formed  at  the  center  of  the  circle 
by  the  radii  is  an  angle  of  90°,  or  a  right  angle. 

7.  How  many  arcs  of  90°  are  there  in  a  circumference  ? 
How  many  right  angles  can  be  formed  at  the  center  of  a 
circle  ? 


289.  Arc  and  Angle  Measure. 

60  seconds  ('')  =  1  minute  (') 
60'  =  1  degree  (°) 
860°  =  a  circumference 
360°  =  4  right  angles  (rt.  -4) 


238 


FORMS  AND   MEASUREMENTS 


29t^.  Latitude  and  Longitude. 

1.  Using  a  map,  point  to  the  equator  ;  to  a  meridian. 
The  equator  is  midway  between  what  two  points?  Do 
the  meridians  extend  around  the  earth  or  only  from  pole  to 
pole  ?  What  is  a  meridian  ?  How  must  two  or  more  places 
be  loc^,ted  to  have  the  same  meridian  ?  How  must  two  or 
more  places  be  located  so  as  not  to  have  the  same  meridians? 
Are  the  meridians  of  all  places  shown  on  the  map  ? 

2.  Places  on  the  earth's  surface  may  be  located  by  two 

measures  taken  from 
two  lines  intersecting 
at  right  angles. 

3.  The  lines  taken 
for  locating  places  on 
a  map  are  the  equa- 
tor and  some  selected 
meridian,  called  the 
prime  meridian.  The 
meridian  of  the  Royal 
Observatory  at  Green- 
wich (near  London), 
England,  is  taken  by 
most  nations  as  the 
prime  meridian. 

4.  The  distance  in  degrees  north  and  south  from  the 
equator  is  called  latitude  ;  and  the  distance  in  degrees  east 
and  west  from  the  prime  meridian  is  called  longitude. 
Places  north  of  the  equator  are  in  north  latitude.  What 
places  are  in  south  latitude  ?  in  east  longitude  ?  in  west 
longitude  ? 

5.  In  the  figure,  point  to  a  place  located  60°  west 
longitude  and  45°  north  latitude  ;  90°  west  longitude  and 


LATITUDE   AND  LONGITUDE  239 

45°  north  latitude  ;  30°  east  longitude  and  45°  south  lati- 
tude ;  45°  west  longitude  and  45°  south  latitude.  Give 
the  latitude  and  longitude  of  each  point  at  which  lines 
intersect  in  the  figure. 

6.  What  is  the  difference  in  degrees  between  two 
places,  one  60°  west  longitude  and  the  other  15°  east 
longitude?  one  105°  west  longitude  and  the  other  30° 
west  longitude  ?  one  45°  north  latitude  and  the  other  60° 
south  latitude  ? 

7.  How  many  degrees  is  it  from  the  equator  to  the 
North  Pole  ?  from  the  equator  to  the  South  Pole  ?  from 
the  North  Pole  to  the  South  Pole  ?  How  many  degrees 
is  it  from  any  point  on  the  equator  halfway  around  the 
earth  ?  one  fourth  way  around  the  earth  ? 

8.  The  equatorial  circumference  of  the  earth  is  24900 
mi.   What  is  the  length  in  miles  of  a  degree  on  the  equator  ? 

9.  What  is  the  latitude  of  your  home  ?  If  the  polar 
circumference  of  the  earth  is  24800  mi.,  how  far  do  you 
live  from  the  equator  ?  from  the  North  Pole  ? 

10.  Which  is  longer,  a  degree  on  the  Arctic  Circle,  on 
the  Tropic  of  Capricorn,  or  on  the  equator  ?     Explain. 

11.  What  is  the  greatest  latitude  that  a  place  can  have  ? 

12.  What  is  the  greatest  longitude  that  a  place  can 
have  either  east  or  west  from  the  prime  meridian  ?    Why  ? 

13.  Where  must  a  place  be  located  to  have  a  latitude  of 
0°  ?  Where  must  a  place  be  located  to  have  a  latitude 
of  0°  and  a  longitude  of  0°  ? 

14.  If  the  prime  meridian  in  the  figure  on  p.  238  is 
the  meridian  of  Greenwich,  locate  on  the  figure  your  own 
home  ;  the  city  of  New  York  ;  Chicago  ;  San  Francisco  ; 
Rio  Janeiro  :  Berlin. 


7 


-A 


240  FORMS  AND  MEASUREMENTS 

291.  Longitude  and  Time  —  Local  Time. 

1.  What  part  of  the  earth's  surface  receives  the  light 
of  the  sun  at  any  one  time  ?  Why  does  the  sun  appear 
to  move  from  east  to  west? 

2.  How  many  hours  does  it  take  the  earth  to  rotate 
once  on  its  axis  ?  Through  how  many  degrees  does  any 
meridian  pass  during  each  rotation  ? 

3.  How  many  degrees  of  longitude  pass  under  the 
sun's  rays  during  24  hours  ?   during  1  hour  ? 

4.  Which  passes  under  the  sun's  rays  first,  the  meridian 
of  your  home  or  that  of  a  place  15°  east  of  you  ?  west  of 
you? 

5.  When  the  vertical  rays  of  the  sun  fall  on  any  part 
of  the  meridian  of  your  home,  it  is  noon  by  sun  time  at 
all  places  on  the  meridian.  Is  it  then  before  noon  or 
after  noon  by  sun  time  at  places  east  of  your  home  ?  west  ? 

6.  Since  the  earth  rotates  through  360°  in  every  24 
hours,  it  must  rotate  through  15°  each  hour.  Therefore 
the  difference  in  sun  time  between  places  15°  apart,  in  an 
east  and  west  line,  is  1  hour. 

7.  When  it  is  noon  by  sun  time  on  the  prime  meridian, 

what  is  the  time  at  a  place  15°  E.?   15°  W.?   30°  E.? 

30°  W.?  45°  W.?  60°  E.?  75°  W.?  90°  W.?  105°  W.? 
^  tr  6  7 

8.  When   it   is  noon   by  sun   time  on  the   principal 

meridian,  what  is  the  longitude  of  a  place  at  which  it 

is   11   A.M.?    1  P.M.?    10  A.M.?   2  P.M.?    9  A.M.?  4  P.M.? 

9.  What  is  the  difference  in  sun  time  between  places 
30°  W.  and  30°  E.  ?  45°  W.  and  60°  E.  ?  30°  W.  and 
60°  W.?  60°  W.  and  105°  W.? 

10.    When  it  is  noon  on  the  prime  meridian,  where  is  it 
midnight?  9  a.m.?  9  p.m.?  6  a.m.?  6  p.m.? 


LONGITUDE  AND  TIME 


941 


292.   Standard  Time. 

To  avoid  the  confusion  that  would  arise  if  every  place 
used  its  own  local  time,  in  1883  the  railroads  of  the 
United  States  and  Canada  agreed  upon  a  system  known 
as  standard  time.  Under  this  system  the  United  States  is 
divided  into  four  time  belts,  each  approximately  15°  in 
width,  and  each  having  the  local  time  of  its  central  me- 
O'idian,  which  is  some  multiple  of  15°.  These  divisions  are 
named  after  the  sections  of  the  country  embraced  by 
them  as  follows :  Eastern,  having  the  time  of  the  meridian 
of  75°  W.  ;  Central,  having  the  time  of  the  meridian  of 
90°  W. ;  Mountain,  having  the  time  of  the  meridian 
105°  W. ;  and  Pacific,  having  the  time  of  the  meridian 
120°  W. 

5  A.M.  6  A.M.  7  A.M.  8  A.M. 


STANDARD  TIME 
BELTS 

105 


While  the  time  belts  are  theoretically  15°  in  width,  they  are 
actually  wider  or  narrower  than  15°.  The  irregularities  of  the 
divisions  are  due  to  the  fact  that  the  railways  find  it  convenient  to  make 
the  changes  in  time  at  the  division  termini  that  are  nearest  to  7|* 
east  or  west  of  the  central  meridians.  ~"  * 


^ 


242  FORMS   AND  MEASUREMENTS 

293.  Map  Questions. 

1.  When  it  is  noon  in  Philadelphia,  what  time  is  it  in 
Chicago?   in  Denver?   in  San  Francisco?   in  New  York? 

2.  When  it  is  9  A.M.  in  Chicago,  what  time  is  it  in 
San  Francisco  ?  in  Washington  ?  in  New  Orleans  ?  in 
Denver?  in  Seattle?  in  New  York? 

3.  A  telegram  was  sent  from  Washington  at  2  p.m. 
and  was  received  in  San  Francisco  at  11  A.M.  of  the  same 
day.     Explain. 

4.  At  10  P.M.  the  people  of  Los  Angeles,  Cal.,  were 
reading  the  election   returns  of   New  York,  which  had 

f  been  compiled  at  11  p.m.  of  the  same  day.     Explain. 

5.  The  passengers  on  a  west-bound  train  arrived  in 
Sparks,  Nev.,  at  6.05  A.M.  and  after  a  stop  of  10  min. 
started  on  their  journey  at  5.15  a.m.     Explain. 

6.  On  leaving  North  Platte,  Neb.,  the  passengers  on  an 
east-bound  train  found  that  their  watches  were  all  1  hr. 
behind  time.     Explain. 

7.  How  many  times  must  a  person  reset  his  watch  in 
traveling  from  Boston  to  San  Francisco,  if  he  wishes  to 
have  correct  time  on  the  journey? 

8.  If  the  telegraph  office  in  Chicago,  111.,  closes  at  6 
p.m.,  what  is  the  latest  time  a  message  can  be  sent  from 
San  Francisco  in  time  to  reach  this  office  before  it  closes, 
allowing  30  min.  for  delays  in  transmission  ? 

9.  When  it  is  noon  by  standard  time  in  western 
Iowa,  is  it  earlier  or  later  than  noon  by  local  time? 
Name  some  place  where  it  is  6  P.M.  by  standard  time 
before  it  is  6  p.m.  by  local  time. 


RATIO  243 

RATIO  ^ 

294.  1.  What  is  the  ratio  of  4  ft.  to  G  ft.  ?  Compare 
the  ratio  of  4  ft.  to  6  ft.  with  the  ratio  of  2  ft.  to  3  ft., 
and  with  the  ratio  of  8  ft.  to  12  ft. 

2.  What  effect  upon  the  ratio  of  two  quantities  has 
(a)  multiplying  both  terms  by  the  same  number  ? 
(6)  dividing  both  terms  by  the  same  number  ? 

3.  Name  two  quantities  whose  ratio  is  -|.  Name  two 
other  quantities  having  the  same  ratio.  ( 

4.  Name  numbers  whose  ratio  is  expressed  by  the 
fraction  f,  f,  |,  |,  f ,  |,  J. 

5.  Name  whole  numbers  whose  ratio  is  as  |  to  2 ; 
as  3  to  J ;  as  |  to  6 ;  as  6  to  |. 

6.  Write  ratios  equivalent  to  the  following,  but  with 
one  or  both  terms  a  fraction :  1  to  2  ;  1  to  4  ;  2  to  3 ;  2 
to  1 ;  3  to  7. 

4      X 

7.  Supply  the  number  in  place  of  x:   -=-— .     The 

fractions  are   stated   as   equivalent   fractions.     Compare 
their  terms  to  find  the  number  in  the  place  of  x. 

8.  The  ratio  between  two  numbers  may  be  stated  in 
the  form  of  a  fraction.  In  |  =  |  we  have  an  equality  of 
ratios.     The  equality  between  ratios  is  called  a  proportion. 

Qi         2_4      6_x      5_10     ^_12     ^_10 
^    ''^-  5~x'  i0~30'   7~  X  '    8~16'    4""l2' 

10.  If  20  bbl.  of  flour  cost  $  80,  how  many  barrels  of 
flour  can  be  bought  for  $120  ?  ($120  is  one  lialf  again  as 
much  as  $80.) 

11.  In  a  certain  city  the  ratio  of  the  number  of  school- 
census  children  to  the  total  population  is  1  to  4J.  If  the 
school  census  is  20,000,  what  is  the  population  of  the  city  ?  . 


/  ; 

PART  V 

POWERS  AND  ROOTS 

295-  3x3  =  9.  3  is  used  twice  as  a  factor  to  give  9. 
9  is  called  the  second  power  of  3.  What  number  is  the 
second  power  of  2  ?  of  4  ?  of  5?  of  10  ?  of  1  ?  of  12  ? 

2.  The  second  power  of  a  number  is  called  its  square,  as 
the  number  of  units  in  the  area  of  a  square  surface  is  found 
by  taking  the  second  power  of  the  number  denoting  the 
length  of  a  side  of  the  square. 

3.  The  square  of  3  may  be  indicated  thus  :  3^.  Indi- 
cate the  square  of  4  ;  of  5  ;  of  1 ;  of  10 ;  of  12.  Give  the 
value  of  each  :  1\  82,  2\  62. 

The  small  figure  written  at  the  right  and  above  indicates 
how  many  times  the  number  is  to  be  taken  as  a  factor  and 
is  called  the  exponent  of  the  number. 

4.  3  X  3.x  3  =  27.  27  is  the  third  power,  or  cube,  of  3. 
What  number  is  the  cube  of  1  ?  of  2  ?  of  4  ?  of  5  ?  of  6  ? 
of  10?   of  12? 

5.  The  cube  of  3  may  be  indicated  by  an  exponent, 
thus  :  33.  Indicate  the  cube  of  7  ;  of  8  ;  of  9.  Give  the 
value  of  each  :  1^,  2^,  lO^,  123. 

5*  is  read  the  fourth  power  of  5,  or  5  to  the  fourth  power ;  it  means 
5x5x5x5.     Read  and  tell  meaning  of :  6*,  3^  2^. 

6.  Find  the  volume  of  a  cube  whose  edge  is  5  in.  Find 
the  cube  of  5. 

7.  Give  the  square  of  each  of  the  numbers  from  1  to  12. 

8.  Square  J,  |,  j,  .5,  1.5,  .04,  ^,  2J. 

9.  Find  and  memorize  the  cubes  of  1,  2, 3, 4, 5,  6, 10, 12. 

244 


POWERS  AND  ROOTS  246 

The  process  of  finding  a  power  of  a  number  is  some- 
times called  involution. 

10.  A  number  that  is  the  square  of  some  integer  or 
fraction  is  called  a  perfect  square.  Thus,  25  (5  x  5)  and 
II  (5  X  I)  are  perfect  squares.     Is  24  a  perfect  square  ? 

11.  Square  each  :  20,  30,  40,  50,  60,  70,  100. 

12.  Is  the  square  of  2  plus  the  square  of  3  the  same  as 
the  square  of  5  ? 

296.  1.  Which  is  the  more  and  how  much,  20^  +  5^  or 
252? 

2.  The  square  of  any  number  composed  of  tens  and 
units  may  be  found  thus  : 

20  +  5  The  square  of  25 

20  +  5  is  seen  to  be  the 

100  +  25     (20  +  5)x5  square  of  the  tens, 

^        '      y  plus     twice     the 

400  +  100  (20  +  5)  X  20  ^..^uct    of    the 

400  +  2(100)  +  25  =  202+  2(20  X  5)  +  52  tens  and  the  units, 

plus  the  square  of 
the  units. 

3.  Square  as  above  :  23,  47,  105  (100  +  5). 

4.  The  figure  represents   a   square   whose  side  is  25 
units.    The  square  whose  side  is  20  units 

contains  400  square  units.     The  two  rec- 
tangles 20  by  5  contain  100  square  units   "^ 
each.     The  square  is  completed  by  the 
addition  of  the  small    square  5  by   5,    o 
containing  25  square  units.     The  area 
of  the  square  is  (400  +  2(20  X  5)  +  25),  ^^ 

or  625  square  units. 


20 

5 

too 

25 

•^ 

400 

100 

0 

5.    Construct  a  square  whose  side  is  10  +  5  units. 


>( 


246  POWERS  AND  ROOTS 

297.  Roots. 

1.  Since  9  is  the  square  of  3,  3  is  the  square  root  of  9 ; 
that  is,  it  is  one  of  the  two  equal  factors  of  9.  What 
number  is  the  square  root  of  4  ?  of  25  ?  of  64  ?  of  36  ?  of 
49  ?  of  16  ?  of  144  ?  of  100  ?  of  81  ?  of  121  ?  of  1  ? 

2.  Since  27  is  the  cube  of  3,  3  is  the  cube  root  of  27. 
What  is  meant  by  the  cube  root  of  a  number?  What 
number  is  the  cube  root  of  1  ?  of  125  ?  of  8  ?  of  1000  ?  of 
1728? 

3.  The  sign  ( V  ')  is  called  the  radical,  or  root  sign,  and 
is  placed  over  a  number  to  show  that  its  root  is  to  be 
taken.  The  root  to  be  taken  is  indicated  by  a  small  figure, 
called  an  index,  written  in  the  radical  thus,  VST,  which 
is  read  the  cube  root  of  27.  The  index  2  for  square  root 
is  usually  omitted. 

4.  Read  and  give  the  roots  :  V64,  "v^,  V49,  VTOO, 
a/125,  V81,  V36,  V144,  -v^. 

The  process  of  finding  the  root  of  a  number  is  some- 
times called  evolution. 

298.  Finding  Roots  by  Factoring. 

Roots  of  perfect  squares  may  be  found  by  factoring. 

1.  Find  the  square  root  of  324. 

By  factoring,  324  =  2  x  2  x  3  x  3  x  3  x  3. 
Arranging  the  factors  into  two  like  groups, 

324  =  (2  X  3  X  3)  X  (2  X  3  X  3). 
V324  =  2  X  3  X  3,  or  18. 

2.  Find  the  cube  root  of  2744. 

Factor  2774.  Group  the  factors  into  three  like  groups.  The 
product  of  one  of  these  groups  is  the  cube  root. 

3.  The  square  root  of  a  fraction  is  the  square  root  of 
its  numerator  over  the  square  root  of  its  denominator, 
thus  :  V|"=  |. 


SQUARE  ROOT  247 

299.    Find  the  roots  indicated :  t^ 

5  4.    ^3875  7.    ^/^OOO        10.    V1296 


Wo  5.    Vl29,600      8.    Vff  11.    V15,625 

3.    Vll,664      6.    ^/'5T2  9.    V^  12.    V6| 

300.  1.  Compare  Vl  =  1,  Vlp  =  10,  and  Vl|00|00 
=  100.  Notice  that  there  is  one  figure  in  the  square  root 
for  each  period  of  two  figures  each  into  which  the  square 
can  be  separated,  beginning  at  units.  The  period  at  the 
left  may  contain  only  one  figure.  By  separating  any 
number  into  such  periods,  the  number  of  figures  in  the 
square  root  may  be  told. 

2.  How  many  figures  are  there  in  the  square  root  of 
each:  11,664?  129,600?  11,025? 

3.  1.22  =  1.44  ;  9.92  =  98.01;  1.222  =  1.4884.  Notice 
that  there  are  two  decimal  places  in  the  square  for  each 
decimal  place  in  the  root. 

4.  How  many  decimal  places  are  there  in  the  square 
root  of  each  :  4.1616  ?  1190.25  ?  2550.25  ? 

301.  Square  Root. 

a.  Find  the  square  root  of  529.  b.  Find  the  side  of  a 
square  whose  area  is  529  square  units. 

As  the  square  root  of  some  numbers  cannot  be  found 
by  factoring,  another  method  of  finding  the  square  root 
of  numbers  is  necessary.  From  Sec.  296  we  see  that  the 
square  of  a  number  is  the  square  of  the  tens,  plus  twice 
the  product  of  the  tens  and  units,  plus  the  square  of  the 
units;  and  from  Sec.  300  we  see  that  the  number  of  fig- 
ures in  the  square  root  of  any  number  is  the  same  as  the 


248 


POWERS  AND   ROOTS 


number  of   periods  of  two  places  each,   beginning  with 
units  into  which  the  number  can  be  separated. 


Model:        5'29|23 
202       =         4 
2x20  =  40     129 
(40+3)  X  3=129 


a.  As  529  can  be  separated  into  two 
periods,  its  square  root  consists  of  tens 
and  units.  Since  the  square  of  tens  is 
hundreds,  5  hundreds  must  include  the 
square  of  the  tens  of  the  root.  The 
largest  perfect  square  in  5  hundreds  is  4 
hundreds.  The  square  root  of  4  hundreds 
is  2  tens.  Write  this  in  the  answer  at  the  right.  The  square  of  2 
tens  is  4  hundreds.  Subtract  4  hundreds  from  529.  The  remainder 
is  129.  This  remainder  must  be  twice  the  product  of  the  tens  and 
the  units,  plus  the  square  of  the  units.  Twice  2  tens  is  40.  The 
units'  figure  of  the  root  is  found  by  taking  40  as  a  partial  divisor. 
40  is  contained  in  120  (omitting  the  9,  as  it  is  evidently  the  square 
of  the  figure  in  units'  place,  or  a  part  of  its  square)  three  times. 
Write  3  as  the  units'  figure  of  the  root.  Use  43  as  the  complete 
divisor.    3  x  43  =  129,  which  exhausts  the  remainder. 


20  B 


60 


A 

AGO 


20 


D3 


h.  As  the  largest  perfect  square  in 
5  hundred  square  units  contains  4 
hundred  square  units,  its  side  is  20 
units  {A).  129  square  units  remain  to 
be  added  in  such  form  as  to  keep  the 
figure  a  square.  It  is  evident  that 
these  units  must  be  added  along  two 
adjacent  sides,  as  B  and  C,  and  at  the 
corner,  as  D.  The  combined  length  of 
the  two  rectangles,  B  and  C,  is  40 
units.  Their  width  may  be  deter- 
mined from  the  fact  that  their  com- 
bined areas,  plus  the  area  of  the  small 
square  D,  is  129  square  units.  Omitting  the  9,  as  it  evidently  is  the 
number  (or  a  part  of  the  number)  of  square  units  in  the  small  square, 
120  square  units  -^  40  square  units  =  3,  the  number  of  units  in 
the  width  of  the  rectangles,  and  also  in  the  side  of  the  small 
square. 


SQUARE   ROOT  249 

302.  To  extract  the  square  root  of  a  number : 

1.  Separate  the  number  into  periods  of  two  figures 
each,  beginning  at  the  decimal  point. 

2.  Find  the  greatest  square  in  the  left-hand  period, 
and  write  its  root  for  the  left-hand  figure  of  the  required 
root. 

3.  Subtract  the  square  from  the  left-hand  period, 
and  bring  down  the  next  period  to  form  the  complete 
dividend. 

4.  Double  the  part  of  the  root  already  found,  and 
place  it  at  the  left  of  the  dividend  for  a  partial  divisor. 
Disregarding  the  right-hand  figure  of  the  dividend,  divide 
by  the  partial  divisor.  The  quotient  (or  quotient  di- 
minished) will  be  the  next  figure  of  the  root. 

5.  Annex  the  root  figure  last  found  to  the  partial  divi- 
sor for  a  complete  divisor.  Multiply  the  complete  divisor 
by  the  root  figure  last  found.  Subtract  the  product  from 
the  dividend,  bring  down  the  next  period  to  form  the 
complete  dividend,  and  continue  as  before. 

303.  Written  Exercises. 

Square  roots  of  numbers  that  are  not  perfect  squares  may  be  ap- 
proximated by  annexing  periods  of  two  decimal  ciphers  and  continu- 
ing the  process  to  several  decimal  places  in  the  roots. 

Extract  the  square  root  of  each  : 

1.    841       /  4    56.25  ^771600 

;2.    104,976  5.    .6724  8.    10.24 

3844  6.    160  U^.  007225 

10.  Find  the  side  of  a  square  whose  area  is  256  sq.  rd. 

11.  Find  the  side  of  a  square  field  whose  area  is  10  A. 

12.  Find  the  perimeter  of  a  square  40-acre  field. 


\ 


250 


POWERS  AND  ROOTS 


D 


304.  1.  Draw  a  right  angle.  Draw  a  rectangle.  Draw 
a  diagonal  through  the  rectangle.  Into  how  many  equal 
triangles  does  the  diagonal  divide  the  rectangle?  What 
kind  of  triangles  are  they  ? 

2.  The  longest  side  of  a  right  triangle  is  called  its 
hypotenuse,  and  the  other  two  sides  are  called  its  legs. 

3.  Draw  a  right  triangle  whose  legs  are  6  in.  and  8  in. 
Measure  the  length  of  the  hypotenuse.  Construct  squares 
upon  each  of  the  three  sides,  and  divide  them  into  square 
inches.  Compare  the  number  of  square  inches  in  the 
square  on  the  hypotenuse  with  the  number  in  the  other 
two  squares  together. 

4.  The  figure  represents  a 
right  triangle  whose  legs  are 
3  units  and  4  units  and  whose 
hypotenuse  is  found  to  be  5 
units.  Compare  the  number 
of  units  in  the  square  upon 
the  hypotenuse  with  the 
number  of  units  in  the  sum 
of  the  squares  upon  the  other 
two  sides. 

The  square  of  the  hypotenuse 
of  a  right  triangle  is  equal  to 
the  sum  of  the  squares  of  the  other  two  sides. 
Answer  the  following  from  the  figure: 

5.  If  the  number  of  squares  in  A  and  B  are  given,  how 
may  the  number  in  C  be  found  ? 

6.  If  the  number  of  squares  in  B  and  C  are  given,  how 
may  the  number  in  A  be  found  ? 

7.  If  the  number  of  squares  in  A  and  C  are  given,  how 
may  the  number  in  B  be  found  ? 


^/<>' 

xyi  3 

ri  " 

B 

A 

-<- 


RIGHT  TRIANGLES 


261 


305.  Written  Exercises. 
1.    Find  the  length  of  the  third  side  of  each 


15ft 


12ft 


20ft  eOrd.  9ft 

2.  Find  the  length  of  the  diagonal  of  the  floor  of  a 
rectangular  room  14  ft.  by  16  ft.,  to  the  nearest  thousandth 
of  a  foot. 

3.  A  boy  stood  on  the  ground  45  ft.  from  the  foot  of 
a  tree  60  ft.  in  height.  How  far  was  it  in  a  straight  line 
from  the  boy's  feet  to  the  top  of  the  tree  ? 

4.  How  much  less  is  the  distance  along  a  diagonal  path 
across  a  rectangular  field  40  rd.  by  80  rd.  than  the  dis- 
tance around  two  sides  of  the  field  ? 

5.  How  long  must  a  rope  be  to  reach  from  the  top  of  a 
60-ft.  pole  to  a  point  on  the  ground  30  ft.  from  the  foot  of 
the  pole  ?  ^ 

6.  Find  the  diagonal  of  a  square  field  whose  side  is 
40  rd. 

7.  Find  the  side  of  a  square  whose  diagonal  is  60  ft. 

8.  Find  the  diagonal  of  a  rectangular  room  20  ft.  by 
26  ft.  If  the  ceiling  of  the  room  is  10  ft.  from  the  floor, 
what  is  the  distance  from  one  of  the  lower  corners  of  the 
room  at  one  end  of  the  diagonal  on  the  floor  to  the  upper 
corner  at  the  other  end  ? 

9.  If  A  is  85  mi.  south  of  B,  and  C  is  75  mi.  west  of 
B,  how  far  is  it  from  A  to  C  ? 

10.  One  side  of  a  rectangular  field  is  40  rd.     The  diag- 
onal is  50  rd.     Find  the  other  side. 

11.  Find  the  diagonal  of  a  10-acre  square  field. 


252  POWERS  AND  ROOTS 

306.  Laying  off  a  Rectangle. 

1.  When  the  two  sides  of  a  rectangle  are  in  the  ratio 
of  3  units  to  4  units,  the  diagonal  is  one  fourth  more  than 
the  longer  side,  thus :  If  the  sides  of  a  rectangle  are  18  ft. 
and  24  ft.,  the  diagonal  is  24  ft.  plus  6  ft.,  or  30  ft. 
Prove  that  this  is  correct  and  that  it  holds  with  various 
rectangles  when  the  ratio  of  the  sides  is  as  3  to  4. 

2.  A  farmer  asked  two  schoolboys  to  lay  off  a  rec- 
tangle 16  ft.  by  24  ft.  to  mark  the  foundation  of  a  car- 
riage house.  The  boys  used  two 
pieces  of  cord  and  a  measure.  They 
tied  the  cords  to  two  stakes  so  that 
they  crossed  at  ^,  a  corner  of  the 
rectangle,  and  extended  one  cord  in 
the  direction  of  cd  and  tlie  other  in 
the  direction  of  ce.     To  make  the 

angle  at  <?  a  right  angle  and  thus  to  get  the  positions  of 
the  two  sides  of  the  rectangle,  they  measured  off  from  c 
15  ft.  on  one  cord  and  20  ft.  on  tlie  other  cord.  With 
these  as  the  legs  of  a  right  triangle,  they  adjusted  the 
position  of  the  cords  so  as  to  make  the  distance  hi  25  ft. 
Having  determined  the  direction  of  the  sides  cs  and  ce^ 
they  measured  24  ft.  on  cs  and  16  ft.  on  ce  and  marked 
the  corners  «  and  i.  From  «  they  extended  a  line  in  the 
direction  so,  and  from  i  a  line  in  the  direction  io.  They 
measured  16  ft.  on  the  line  so  and  24  ft.  on  the  line  io. 
They  marked  the  point  o  where  the  two  measured  lines 
met.  They  tested  their  work,  finding  that  the  diagonals 
CO  and  si  were  equal. 

3.  Using  cords  and  a  measure,  lay  out  on  the  school 
grounds  rectangles  12  ft.  by  16  ft. ;  20  ft.  by  30  ft. 

4.  Lay  off  a  baseball  "  diamond  "  whose  side  is  60  ft. 


SIMILAR  FIGURES  268 

SIMILAR  SURFACES   AND   SOLIDS 

307,  1.  Draw  squares  whose  sides  are  1  incli;  2  inches; 
3  inches.  Find  their  areas.  The  areas  of  the  three  squares 
are  to  each  other  as  1^,  2  2,  and  3  2. 

2.  Express  the  ratio  of  the  areas  of  a  4-inch  square  and 
a  6-inch  square. 

3.  Draw  circles  whose  diameters  are  4  inches ;  6  inches. 
As  the  area  of  a  circle  is  Trr^,  the  ratio  of  the  areas  of  these 
two  circles  is  as  2^  is  to  3  2.     Explain. 

4.  Is  the  ratio  of  the  squares  of  the  diameters  of  two 
circles  the  same  as  the  ratio  of  the  squares  of  their  radii  ? 

5.  Is  the  ratio  of  the  squares  of  the  diagonals  of  two 
squares  the  same  as  the  ratio  of  the  squares  of  their  sides  ? 

6.  Figures  that  are  of  exactly  the  same  shape  are  called 
similar  figures.  Draw  two  similar  figures.  Are  similar 
figures  necessarily  the  same  in  size  ? 

The  areas  of  similar  plane  figures  are  proportional  to  the 
squares  of  their  corresponding  lines, 

7.  Find  the  volume  of  a  2-inch  cube  ;  of  a  3-inch  cube. 
Their  volumes  are  in  the  ratio  of  2^  to  3  3. 

The  volumes  of  similar  solids  are  proportional  to  the  cubes 
of  their  corresponding  lines, 

8.  Compare  the  volumes  of  two  spheres,  one  5  inches  in 
diameter  and  the  other  10  inches  in  diameter. 

As  the  diameter  of  the  larger  sphere  is  twice  the  diameter  of  the 
smaller,  the  volume  of  the  larger  is  2*  times  the  volume  of  the  smaller. 
Explain. 

9.  Compare  the  weights  of  two  solid  iron  spheres  of  the 
same  density,  if  one  is  2  inches  in  diameter  and  the  other 
is  4  inches  in  diameter. 


254  MISCELLANEOUS  EXERCISES 

MISCELLANEOUS   EXERCISES 

308.  1.  A  merchant  gained  -f  28  by  selling  some  goods 
at  a  profit  of  20  % .     Find  the  cost  of  the  goods. 

2.  An  agent  who  canvassed  for  a  book  received  40% 
of  the  amount  of  the  sales  for  selling  and  delivering  the 
books.  Find  the  amount  of  his  commission  for  selling 
and  delivering  60  copies  at  11.25  per  copy. 

3.  If  the  amount  received  for  goods  sold  averages 
20  %  more  than  the  cost  of  the  goods,  find  the  net  profit 
for  a  month  when  the  sales  amounted  to  $24,000  and  the 
expenses  for  the  month  amounted  to  f  3000. 

4.  20  %  of  the  pupils  enrolled  in  a  certain  school  were 
absent  one  stormy  day.  Twenty-four  pupils  were  present. 
Find  the  number  enrolled. 

5.  A  commission  merchant  sold  $6000  worth  of  prod- 
uce at  a  commission  of  1|%.  Find  the  amount  of  his 
commission. 

6.  If  a  collector  received  20  %  commission  for  collect- 
ing a  bill  of  $17.75,  what  was  the  amount  of  his  commis- 
sion? What  per  cent  of  the  amount  of  the  bill  did  the 
creditor  receive? 

7.  A  man  paid  $42  taxes  when  the  tax  rate  was  2%. 
What  was  the  assessed  valuation  of  his  property  ? 

8.  If  a  person  pays  $50  tax  on  property  when  the  rate 
is  1-|  %,  what  is  the  assessed  valuation  of  his  property  ? 

9.  A  miller  bought  a  ton  of  wheat  through  a  broker, 
who  charged  a  commission  of  2  %.  What  was  the  amount 
paid  for  the  wheat  if  the  cost  of  the  wheat  and  the  broker- 
age amounted  to  $25.50  ? 

The  cost  of  the  wheat  plus  2%  of  the  cost  of  the  wheat,  or  102%  of 
the  cost  of  the  wheat,  was  $25.50.     Prove  your  answer. 


MISCELLANEOUS  EXERCISES  265 

10.  A  fruit  grower  shipped  25  boxes  of  apples  to  a 
commission  merchant,  who  sold  them  at  85^  per  box, 
charging  4  %  commission.  He  was  directed  to  invest  the 
proceeds  in  groceries,  after  deducting  a  commission  of  2  % 
for  making  the  purchase.  Find  the  amount  expended 
for  the  groceries. 

The  net  proceeds  of  the  sale  of  the  apples  was  $20.40,  which  was 
102  7o  of  the  amount  invested  in  groceries.     Prove  your  answer. 

11.  After  selling  25  %  of  his  interest  in  a  flour  mill,  a 
man  considered  his  remaining  interest  worth  $  9000.  At 
this  rate,  what  was  the  value  of  his  interest  before  making 
the  sale  ? 

12.  A  real  estate  broker  bought  two  60-ft.  lots,  adjoin- 
ing, for  $  1500  apiece,  and  divided  them  into  three  lots  of 
equal  frontage  which  he  sold  for  #1200  apiece.  What 
per  cent  of  profit  did  he  make? 

-    13.    At  $  9  a  ton,  find  the  cost  of  two  sacks  of  coal,  each 
weighing  100  lb. 

14.  A  man  bought  a  lot  for  $  1200  and  sold  it  for  1 1400. 
He  bought  it  back  for  $1500  and  resold  it  for  $1600. 
How  much  did  he  make  on  the  lot  ? 

15.  A  man  bought  four  50-ft.  lots  and  divided  the  land 
into  40-ft.  lots,  which  he  sold  at  the  same  price  per  lot  as 
he  had  paid.     Find  his  gain  per  cent. 

16.  At  22  ^  a  square  foot,  find  the  cost  per  front  foot  of 
paving  a  street  60  ft.  wide.  Find  the  cost  per  front  foot 
to  a  property  owner  who  pays  for  half  the  width  of  the 
street.  Find  the  cost  to  the  property  owner  of  paving  the 
street  in  front  of  a  45-ft.  lot. 

17.  If  27  tons  of  coal  cost  1 243,  how  many  tons  can  be 
bought  for  1 189  ? 


PART   VI 

APPENDIX 
CORPORATIONS,   STOCKS,  AND  BONDS 

309.  1.  Corporations.  A  large  business  enterprise  frequently 
requires  more  capital  than  one  person  may  care  to  invest  in 
it.  Provision  is  made  in  the  laws  of  the  various  states 
whereby  a  number  of  persons  may  organize  a  company,  called 
a  corporation,  to  engage  in  business  as  one  body.  Sometimes 
all  the  necessary  capital  is  subscribed  by  the  persons  who  organ- 
ize the  corporation,  but  often  the  organizers  of  a  company  sub- 
scribe only  a  part  of  the  capital. 

The  laws  regulating  the  incorporation  of  companies  differ  considerably 
in  the  several  states.  Frequently  a  corporation  intending  to  transact  busi- 
ness in  one  state  will  incorporate  in  another  state,  because  of  certain  ad- 
vantages to  be  derived  thereby. 

2.  Railway  companies,  mining  companies,  express  com- 
panies, oil  companies,  and  banking  institutions  are  among  the 
largest  business  corporations. 

310.  Shares  of  Stock.  1.  Each  corporation  is  capitalized  for 
a  special  amount,  as  $  25,000,  $  50,000,  $  1,000,000,  etc.  The 
capital  is  divided  into  shares,  usually  of  $  100  each  or  of  $1 
each.  Thus,  a  corporation  that  is  capitalized  for  $  100,000  may 
issue  1000  shares  of  the  face  value  of  $  100  each,  or  100,000 
shares  of  the  face  value  of  $  1  each,  etc.  These  shares  are 
bought  by  persons  who  invest  in  the  enterprise.  Each  person 
who  owns  one  or  more  shares  of  stock  is  called  a  stockholder. 
The  several  stockholders  constitute  the  corporation. 

266 


CORPORATIONS,   STOCKS,   AND   BONDS  257 

2.  Every  stockholder  receives  a  certificate  of  stock,  showing 
the  number  of  shares  he  owns  and  the  face  value,  or  par 
value,  of  each.  These  certificates  are  negotiable,  and  a 
record  of  their  transfer  is  usually  made  on  the  books  of  the 
corporation. 

The  affairs  of  a  corporation  are  managed  through  a  board  of  directors, 
elected  by  the  stockholders,  each  stockholder  having  as  many  votes  as  the 
number  of  shares  of  stock  he  owns. 

311.  Value  of  Stock,  l.  The  price  at  which  stocks  are 
bought  and  sold  in  the  stock  market  is  called  their  market  value. 
When  the  market  value  of  stock  is  the  same  as  its  face  value, 
the  stock  is  said  to  be  at  par.  Stock  is  said  to  be  at  a  premium^ 
or  above  par,  when  its  market  value  is  more  than  its  face  value, 
and  at  a  discount,  or  below  par,  when  its  market  value  is  less 
than  its  face  value. 

2.  Examine  the  stock  quotations  in  a  newspaper.  Can  you 
tell  from  the  quotations  what  the  face  value  of  the  stock  is  ? 
Tell  which  stock  is  at  par,  above  par,  below  par. 

312,  Dividends.  The  net  earnings  of  a  corporation,  after 
a  surplus  sufficient  to  cover  the  probable  needs  has  been  re- 
served, are  divided  among  the  stockholders  according  to  the 
number  of  shares  owned  by  each.  These  divided  profits  are 
called  dividends.  Dividends  are  computed  on  the  par  value  of 
the  stock,  and  are  declared  annually,  semiannually,  quarterly, 
etc. 

Illustration.  If  the  amount  of  capital  stock  is  $  500,000 
and  the  amount  to  be  divided  among  the  stockholders  is 
$25,000,  the  rate  of  dividend  is  $  25,000 --$ 500,000,  or  5%. 
A  stockholder  who  owns  stock  of  the  face  value  of  $10,000 
will  receive  5%  of  $10,000,  or  $500. 

Stock  that  regularly  pays  a  large  dividend  is  usually  regarded  as  a  good 
investment,  and  is  likely  to  be  above  par.  When  the  dividends  are  not 
equivalent  to  a  fair  rate  of  interest  on  the  investment,  the  stock  is  likely 
to  be  below  par. 

MccL.  &  Jones's  essen.  of  ar.  — 17 


258  APPENDIX 

313.  Corporations  sometimes  issue  two  kinds  of  stock,  called 
preferred  and  common.  When  both  kinds  of  stock  are  issued, 
the  holders  of  common  stock  are  not  entitled  to  participate  in 
the  profits  until  a  fixed  rate  of  dividend  has  been  paid  to 
holders  of  preferred  stock. 

314.  Stock  Brokers.  1.  A  person  who  is  engaged  in  buying 
and  selling  stocks  for  others  is  called  a  stock  broker.  Stocks  are 
usually  bought  and  sold  through  brokers,  generally  at  a  regu- 
lar meeting  place  for  transacting  such  business,  called  a  stock 
exchange. 

2.  The  commission  of  a  broker  is  called  brokerage.  The  rate 
of  brokerage  varies  in  different  parts  of  the  country  from 
1%  of  the  par  value  for  buying  and  also  for  selling,  to  ^%  or 
more.  A  minimum  amount  is  fixed  for  making  small  sales 
and  purchases. 

The  standard  amount  of  stock  bought  and  sold  is  100  shares,  although 
a  smaller  amount  may  be  negotiated.  As  a  rule,  fractions  of  a  share 
cannot  be  bought. 

In  stock  quotations  fractions  are  always  expressed  in  halves,  fourths, 
and  eighths.     See  quotations,  p.  259. 

315.  Assessments.  When  the  funds  of  a  corporation  are  not 
sufficient  to  carry  on  its  business,  an  assessment  is  sometimes 
levied  upon  the  stock.  The  assessment  is  usually  some  number 
of  cents  per  share,  and  the  failure  to  pay  it  is  generally  pun- 
ishable by  the  forfeiture  of  the  stock. 

316.  Examine  a  newspaper  for  stock  quotations.  What 
is  meant  by  par  value  ?  premium  ?  dividend  ?  market  value  ? 
brokerage  ?  Name  some  of  the  corporations  engaged  in  busi- 
ness in  the  state  in  which  you  live.  How  many  shares  of 
stock  of  the  par  value  of  $  100  each  are  issued  by  a  corporation 
tliat  is  capitalized  for  ^  500,000  ?  for  $  2,000,000 ?  What  is  a 
stockholder  ? 


CORPORATIONS,   STOCKS,   AND   BONDS  259 

Stock   Quotations 


317,     New  York  Stock 

San  Francisco  Stock  . 

A.ND 

Exchange 

Exchange  Board 

Mar.  9, 
190T 

Jan.  22, 
1907 

Mar.  9, 
190T 
Mining 

Jan.  22, 
1907 

Adams  Express 

295 

300 

Caledonia 

.43 

.67 

American  Express      223 

242 

Confidence 

1.05 

1.30 

C.  &  N.  W. 

154| 

190^ 

Combin.  Frac. 

4.45 

5.50 

Denver  &  Rio  G. 

31f 

39^ 

Jumbo  Exten. 

1.85 

1.90 

Denver  &  Rio  G. , 

pfd.  72 

81 

Mustang 

.27 

.21 

Illinois  Central 

147| 

166 

Ophir 

2.65 

3.10 

Northern  Pacific 

137 1 

155^ 

Red  Top  Exten. 

.50 

.37 

Southern  Pacific 

84| 

95| 

Silver  Pick 

1.35 

1.45 

Pullman  Car  Co. 

165 

173| 

St.  Ives 

1.82 

.93 

Union  Pacific 

1561 

176| 

Utah 

.06 

.08 

Do.  preferred 

88 

921 

Vernal 

.20 

.25 

Wells  Fargo  Ex. 

280 

300 

West  End 

1.40 

1.90 

Western  Union 

80f 

83| 

Yellow  Jacket 

1.05 

1.15 

The  par  value  of  the  stock  quoted  in  the  left-hand  column  is  $100  per 
share,  and  in  the  other  column  is  $1  per  share.  On  Jan.  22  the  mar- 
ket was  high,  and  on  March  9  low.  Which  of  the  stocks  quoted  are  above 
par  ?  below  par  ? 

At  the  above  prices,  find  the  cost  without  brokerage,  on 
March  9,  of  100  shares  of  Wells  Fargo  Express  stock;  of 
100  shares  of  Chicago  and  Northwestern  Railway  stock  ;  of  100 
shares  of  Western  Union  stock ;  of  100  shares  of  stock  in  the 
Ophir  mine;  of  100  shares  of  stock  in  the  Silver  Pick  mine. 
Find  the  same  for  the  prices  given  for  Jan.  22. 

318.   Written  Exercises. 

1.  Find  the  cost  of  100  shares  of  Western  Union  stock  at 
30|,  including  \  %  brokerage. 

Market  price  of  each  share,  $80|. 

Brokerage  on  each  share,  ^     \. 

Cost  of  each  share  (including  brokerage),  $80|,    or  $80.76. 

Cost  of  100  shares  (including  brokerage),  $8075. 


260  APPENDIX 

2.  Find  the  cost  of  100  shares  of  Northern  Pacific  stock  as 
quoted  for  March  9,  including  |  %  brokerage. 

3.  Find  the  net  proceeds  of  the  sale  of  100  shares  of  Southern 
Pacific  stock  as  quoted  for  Jan.  22,  allowing  |%  brokerage. 

Market  price  of  each  share,  $95f. 

Brokerage  on  each  share,  $     J. 

Net  proceeds  on  each  share,  $95|,      or  $95.25. 

Net  proceeds  on  100  shares,  $  9525. 

4.  Find  the  net  proceeds  from  the  sales  of  400  shares  of 
Illinois  Central  stock  as  quoted  for  March  9,  allowing  J% 
brokerage. 

5.  How  much  would  a  man  clear  by  buying  100  shares  of 
stock  at  142^  and  selling  them  at  150|,  allowing  ^  %  broker- 
age both  for  buying  and  for  selling  ? 

6.  If  the  Pullman  Car  Company  declares  a  dividend  of  10  %, 
how  much  will  a  person  receive  who  owns  100  shares  of  the 
stock  ? 

Par  value  of  100  shares,  $10,000.  His  dividend  will  amount  to  10%  of 
$10,000. 

7.  If  the  Union  Pacific  Railway  declared  a  dividend  of 
8%,  how  much  did  a  person  receive  who  owned  500  shares  of 
the  stock  ? 

8.  Find  the  cost  of  100  shares  of  stock  at  $1.20  a  share,  in- 
cluding J  %  brokerage.  If  a  10  %  dividend  is  declared  on  this 
stock,  how  much  does  a  person  receive  who  owns  100  shares  of 
the  stock  ? 

9.  What  per  cent  does  a  person  receive  on  the  amount  in- 
vested who  buys  100  shares  of  stock  at  $1.20,  paying  ^  % 
brokerage,  and  receives  a  dividend  of  10  %  ? 

The  dividend  amounts  to  10%  of  $  100.  This  amount  is  what  per  ceot 
of  $  120.50,  the  cost  of  the  stock  ? 


CORPORATIONS,   STOCKS,   AND  BONDS  261 

10.  A  man  bought  100  shares  of  Bed  Top  Extension  stock 
at  37  cents,  paying  $  1  brokerage.  Find  the  cost  of  the  stock. 
If  he  received  a  dividend  of  5  %,  what  per  cent  did  he  receive 
on  his  investment  ? 

$5,  the  dividend,  is  what  per  cent  of  $38,  the  amount  invested  ? 

11.  What  per  cent  did  a  man  make  on  his  investment  who 
bought  100  shares  of  stock  at  45  ^  and  sold  them  at  55/,  pay- 
ing $  1  brokerage  both  for  buying  and  for  selling  ? 

Cost  of  the  shares,  $46;  received  for  the  shares,  $64;  net  profit,  $9. 
$9  is  what  per  cent  of  $46  ? 

319.  Corporation  Bonds.  The  promissory  note  of  a  corpora- 
tion, issued  under  seal,  is  called  a  bond.  The  bond  of  a  cor- 
poration is  secured  by  a  mortgage. 

320.  A  city  or  an  incorporated  village  is  called  a  municipal 
corporation. 

321.  1.  Governments,  states,  cities,  counties,  etc.,  are  some- 
times obliged  to  issue  bonds  to  meet  urgent  demands  for  money 
and  to  provide  needed  improvements.  Such  bonds  are  not 
secured  by  mortgages.  The  integrity  of  the  government 
issuing  a  bond  is  accepted  as  sufficient  security  for  its  pay- 
ment. The  bond  of  a  government  is  a  certificate  of  in- 
debtedness, with  a  promise  to  pay  a  certain  sum  to  the  holder 
of  the  bond,  with  a  fixed  rate  of  interest  at  a  specified 
time,  as  at  the  expiration  of  five  years,  twenty  years,  fifty 
years,  etc. 

2.  Under  what  conditions  is  it  sometimes  necessary  for  the 
United  States  government  to  borrow  money  ?  For  what  pur- 
poses are  cities  frequently  bonded  ?  What  is  a  bond  election  ? 
Which  is  the  better  security,  the  note  of  an  individual  or  the 
bond  of  the  national  government  ? 

322.  Bonds  that  are  registered  by  number  and  in  the  name 
of  the  holder  are  called  registered  bonds.  Bonds  having  interest 
certificates  attached  in  the  form  of  coupons  are  called  coupon 
bonds. 


262  APPENDIX 

Bond  Quotations 

323.  Examine  a  newspaper  for  bond  quotations.  The  fol- 
lowing quotations  are  from  a  newspaper  report  of  prices  on 
the  New  York  Stock  Exchange. 

Japan  6's 99|      Southern  Pacific  4's    .    .    84 

Mexican  Central  4's  .     83      U.  S.  New  4's  reg. .     .    .  129^ 
Northern  Pacific  4's  .  100|  Do.  coupon    ....  129^ 

The  par  value  of  the  bonds  quoted  is  $100  each.  Bonds  are  bought 
and  sold  in  the  market  in  the  same  manner  as  stocks.  Government 
bonds  are  exempt  from  taxation. 

Find  the  cost  of  Mexican  Central  bonds  of  the  face  value  of 
$1000,  including  |  %  brokerage. 

COMMISSION  AND  BROKERAGE 

324,  Producer  and  Consumer.  A  person  who  grows  agri- 
cultural products  or  manufactures  useful  articles  out  of  crude 
materials' is  called  a  producer.  A  person  who  uses  up  products 
is  called  a  consumer.  Name  some  producers  of  foods,  of  cloth- 
ing, of  fuel,  of  building  materials.  Name  some  products  that 
are  consumed  by  nearly  every  one.  Name  some  classes  of 
persons  who  are  consumers  but  are  not  producers. 

In  early  times  there  was  comparatively  little  buying  and  selling,  and 
the  exchange  of  products  was  very  limited.  Each  family  produced  nearly 
everything  that  it  consumed.  At  that  time,  most  of  the  trade  was  directly 
between  the  producer  and  the  consumer.  With  the  invention  of  machin- 
ery and  with  improved  means  of  transportation,  cities  increased  rapidly 
in  number  as  centers  of  manufacture  and  trade.  People  began  to  devote 
themselves  more  particularly  to  special  lines  of  work.  As  trade  con- 
ditions grew  more  complex,  it  became  more  difficult  for  the  producer  to 
trade  directly  with  the  consumer.  Wholesale  and  retail  establishments 
developed  as  agencies  for  marketing  products. 

Middleman.  A  person  who  deals  between  the  producer  and  the  con- 
sumer is  known  in  trade  as  a  middleman.  Products  are  often  handled  by 
several  middlemen  before  they  reach  the  consumers.  The  middlemen  are 
generally  persons  who  make  buying  and  selling  products  their  special 
occupation.    Is  the  retail  dealer  a  middleman  ?    Which  of  the  following 


COMMISSION  AND  BROKERAGE  263 

are  middlemen  :  farmers  ?  stock  buyers  ?  hardware  merchants  ?  carpen- 
ters ?  shoemakers  ?  hay  and  grain  dealers  ?  What  are  some  of  the  con- 
ditions that  make  it  inconvenient  for  persons  living  in  large  cities  to  buy 
agricultural  products  directly  from  farmers?  What  are  some  of  the 
conditions  that  make  it  inconvenient  for  farmers  to  buy  their  clothing  and 
tools  directly  from  the  manufacturers  ?  Which  contributes  to  the  vs^ealth 
of  a  country :  the  producer,  the  consumer,  or  the  middleman  ? 

325.  Commission.  A  person  who  transacts  business  for  an- 
other frequently  receives  as  his  pay  a  certain  rate  per  cent  on  the 
amount  involved  in  the  transaction.  This  is  known  as  his  com- 
mission. One  who  buys  or  sells  produce  for  another,  receiving  as 
his  pay  a  certain  rate  per  cent  on  the  cost  of  the  products  bought 
or  on  the  selling  price  of  the  products  sold,  is  called  a  com- 
mission merchant.  A  commission  house  is  an  establishment  con- 
ducted by  a  commission  merchant,  where  products  are  received 
and  sold  to  retail  dealers  or  consumers.  Why  are  the  commission 
houses  located  in  the  cities  ?  If  your  home  is  in  a  city  in  which 
there  are  commission  houses,  tell  in  what  section  of  the  city  they 
are  located.  If  your  home  is  in  the  country,  tell  what  products 
are  shipped  from  your  community  to  commission  merchants. 

Farmers  frequently  dispose  of  their  products  by  shipping  them  to 
commission  merchants  in  the  cities,  who  sell  the  products  at  the  market 
prices  and  retain  as  their  pay  a  certain  per  cent  of  the  selling  price. 
Commission  merchants  do  not  usually  buy  the  products  shipped  to  them, 
but  they  act  merely  as  the  agents  of  the  shippers  in  receiving  and  selling 
the  goods.  The  freight  charges  for  shipping  and  the  cartage  charges  for 
hauling  the  goods  are  generally  paid  by  the  commission  merchants  from 
the  proceeds  of  the  sale  of  the  products.  After  deducting  their  commis- 
sion and  the  charges  for  freight  and  cartage,  the  commission  merchants 
remit  to  the  shippers  the  balance  of  the  sum  received  for  the  products  sold. 

The  entire  amount  received  from  the  sale  of  goods,  before  any  deduc- 
tion is  made  for  expenses,  etc.,  is  called  the  gross  receipts  of  the  sale. 
The  amount  remaining  from  the  sale  of  goods  after  all  expenses  have  been 
deducted  is  called  the  net  receipts  of  the  sale. 

Where  do  city  retail  dealers  buy  their  supplies  of  fruits,  vegetables, 
etc.  ?  Farmers  frequently  sell  or  exchange  small  quantities  of  agricul- 
tural products  at  the  general  merchandise  stores  in  small  cities  and  villages. 
How  do  these  merchants  dispose  of  the  products  ? 


264  APPENDIX 

326.  Market  Reports.  Newspapers  publish  market  reports^ 
giving  the  prices  at  which  grain,  live  stock,  dairy  products, 
fruits,  etc.,  were  sold  on  the  previous  day.  Of  what  use  are 
such  reports  ?  Read  a  recent  market  report.  The  price  of 
produce  is  affected  by  the  supply  offered  for  sale  and  the 
demand  for  it.  What  effect  upon  prices  has  an  increase  in 
supply  and  a  decrease  in  demand  ?  How  is  the  price  affected 
by  an  increase  in  demand  and  a  decrease  in  supply  ? 

327.  Brokerage.  1.  One  who  acts  as  an  agent  for  others  to 
contract  for  the  purchase  or  sale  of  goods,  receiving  as  his 
pay  a  certain  rate  of  commission,  is  called  a  broker.  The  com- 
mission of  a  broker  is  called  brokerage.  Commission  mer- 
chants usually  take  possession  of  the  goods  bought  and  sold 
by  them,  while  brokers  merely  contract  for  the  sale  or  purchase 
of  goods  in  the  name  of  the  person  buying  or  selling,  without 
taking  possession  of  the  goods.  Brokers  deal  in  stocks,  bonds, 
grain,  etc. 

2.  What  is  real  estate  ?  One  who  buys  and  sells  lands,  ex- 
changes and  leases  property,  etc.,  is  called  a  real  estate  agent, 
or  a  real  estate  broker.  He  usually  receives  as  his  commission 
a  certain  rate  per  cent  on  the  selling  price  of  property,  or  on  a 
month's  rent  when  property  is  leased  by  the  month.  The 
rates  of  commission  charged  for  selling  and  renting  property 
vary  in  different  communities.  Find  what  rate  of  commission 
is  charged  by  agents  for  selling  and  renting  property  in  your 
community.  What  are  some  of  the  conditions  that  make  it 
inconvenient  for  each  person  to  sell  or  rent  his  own  property  ? 
When  a  person  wishes  to  rent  a  house  in  a  city,  how  does  he 
find  out  what  houses  are  for  rent  ? 

3.  Traveling  salesmen,  store  clerks,  auctioneers,  insurance 
agents,  etc.,  frequently  receive  as  pay  for  their  services  a  com- 
mission on  the  amounts  of  their  sales.  Can  you  name  any 
other  business  in  which  those  employed  receive  a  commission 
for  their  services  ? 


TRADE   DISCOUNT  265 

TRADE   DISCOUNT 

328.  1.  Manufacturers  and  wholesale  dealers  issue  cata- 
logues and  price  lists  in  which  the  articles  manufactured  by 
them  are  described  and  their  prices  given.  Most  manufacturers 
sell  their  goods  to  wholesale  dealers,  who  supply,  in  turn,  the 
retail  dealers.  The  prices  quoted  in  catalogues  and  price  lists 
are  commonly  known  as  the  list  prices  of  goods.  A  discount 
from  the  list  prices  is  generally  made  to  retail  dealers,  and 
sometimes  to  others  when  goods  are  bought  in  large  quanti- 
ties. This  discount  is  reckoned  as  a  certain  rate  per  cent  of 
the  list  price.  Such  a  discount  is  generally  known  as  trade 
discount. 

2.  Often  several  discounts  are  allowed.  Thus,  an  article 
may  be  sold  subject  to  discounts  of  25  %,  10  %,  and  5  %  from 
the  list  price.  In  such  a  case,  the  first  discount  is  from  the  list 
price,  and  the  second  discount  is  from  the  price  after  deducting 
the  first  discount,  and  the  third  discount  is  from  the  price 
after  deducting  the  second  discount.  Usually  the  amount  of 
discount  allowed  is  varied  as  the  market  prices  change. 

3.  A  special  discount  is  usually  made  when  cash  is  paid  for 
goods.  The  amount  of  cash  discount  varies  considerably  in 
different  lines  of  trade,  ranging  generally  from  i  %  to  6  %  of 
the  amount  of  the  bill  after  deducting  the  trade  discount,  and 
averaging  about  2  %.  Generally  the  retail  dealer  is  given 
until  about  the  tenth  day  of  the  following  month  in  which  to 
make  cash  remittances.  Instead  of  allowing  a  special  discount 
for  cash  payments,  sometimes  an  arrangement  is  made  by 
which  the  retail  purchaser  is  given  30  days,  60  days,  90  days, 
6  months,  or  even  a  longer  time  in  which  to  make  his  payment. 
The  bill  then  becomes  due  at  the  end  of  the  specified  time,  and 
in  case  it  is  not  paid  when  the  time  has  expired,  the  purchaser 
of  the  goods  agrees  to  pay  a  specified  rate  of  interest  upon 
the  amount  of  the  bill  from  the  time  the  bill  is  due  until  it  is 
paid. 


266  APPENDIX 


PARTIAL  PAYMENTS 

329,  Instead  of  paying  the  whole  amount  of  a  note  at  one 
time,  the  maker  sometimes  pays  it  in  two  or  more  parts.  Such 
payments  are  called  partial  payments.  A  record  of  each  pay- 
ment is  indorsed  on  the  back  of  the  note. 

330.  United  States  Rule.  The  method  of  computing  in- 
debtedness when  partial  payments  have  been  made,  illustrated 
in  the  following  problem,  was  adopted  by  the  Supreme  Court 
of  the  United  States,  and  is  commonly  known  as  the  United 
States  Rule.     This  method  is  the  legal  one  in  most  states. 

In  states  where  other  methods  are  legal,  teachers  should  follow  thera. 

1.  A  note  for  $  2000  dated  May  1, 1906,  at  6%,  was  indorsed 
as  follows:  July  25,  1906,  $150;  Dec.  16,  1906,  $40;  Feb.  12, 
1907,  $  100.     Find  the  amount  due  May  1,  1907. 

Principal,  May  1,  190a ^2000 

Interest  on  $  2000  to  July  25  (2  mo.  24  da.)  .     .     .     .  28 

Amount,  July  25,  1906 $2028 

First  payment  (July  25,  1906) 150 

New  Principal,  July  25,  1906 §  1878 

Interest  on  $  1878  to  Dec.  16,  1906  (4  mo.  21  da.)      .  44.13 

Second  payment,  which  is  less  than  the  interest  due, 

$40 

Interest  on  $  1878  from  Dec.  16,  1906,  to  Feb.  12, 1907 

(1  mo.  26  da.) 17.53 

Amount,  Feb.  12,  1907 $  1939.66 

Third  payment,  $100,  which  is  to  be  added  to  the 

second,  $40 140 

New  Principal,  Feb.  12,  1907 §  1799.66 

Interest  on  $1799.66  to  May  1,  1907  (2  mo.  19  da.)     .  23.69 

Amount  due  May  1,  1907 $1823.35 

If  the  second  payment,  $40,  which  was  less  than  the  interest  due,  had 
been  deducted  from  the  amount  due  at  the  time  the  payment  was  made, 
and  if  the  remainder  had  been  regarded  as  a  new  principal,  the  effect 
would  have  been  to  increase  the  amount  on  which  interest  was  paid. 
Hence,  the  interest  must  be  reckoned  to  the  time  of  the  third  payment. 


INTEREST 


267 


BuLE.  Fi7id  the  amount  of  the  principal  to  a  time  when  a 
payment,  or  the  sum  of  two  or  more  payments,  equals  or  exceeds 
the  interest  due. 

Subtract  the  payment  or  payments  from  the  amount. 

Treat  the  remainder  as  a  new  principal  and  proceed  as  before. 

2.  Write  a  note  for  $  800,  naming  the  teacher  as  payee  and 
yourself  as  maker.  Make  three  partial  payments  and  have 
them  indorsed  on  the  note.  Find  the  amount  due  at  the  time 
of  settlement. 

3.  Write  a  note  for  $  1000,  naming  some  pupil  as  payee  and 
yourself  as  maker.  Make  two  payments,  such  that  the  first  is 
less  than  the  interest  to  date,  but  the  sum  of  both  exceeds  the 
interest  to  time  of  the  second  payment.  Find  the  amount  due 
at  time  of  settlement. 

INTEREST 

331.  Bankers'  Table  of  Days  Betweex  Dates 


Jan. 

Feb. 

Mar. 

Apr. 

May 

June 

JUL^ 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

Jan. 

365 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

Feb. 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

Mar. 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

Apr. 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May 

245 

276 

304 

335 

365 

31 

61 

92 

123 

163 

184 

214 

June 

214 

245 

273 

304 

334 

365 

30 

61 

92 

122 

153 

183 

July 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

Aug. 

153 

184 

212 

243 

273 

304 

334 

365 

31 

61 

92 

122 

Sept. 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

Oct. 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

Nov. 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

Dec. 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

The  exact  number  of  days  from  any  day  of  one  month  to  the  same 
day  of  another  month  within  a  year  is  found  thus :  Find  the  number  of 
days  from  April  12  to  Aug.  12.  Starting  from  April,  in  the  left-hand  col- 
unm,  pass  the  pencil  across  to  the  column  headed  August.  The  number 
122  in  the  column  headed  August  denotes  the  number  of  days  from  any 
day  in  April  to  the  corresponding  day  in  August.  Hence  it  is  122  days 
from  April  12  to  Aug.  12. 


268  APPENDIX 

1.  How  many  days  must  be  added  to  122  days  if  you  are 
required  to  find  the  number  of  days  from  April  12  to  Aug. 

14  ?  to  Aug.  20  ?  How  many  days  must  be  subtracted  from 
122  days  if  you  are  required  to  find  the  number  of  days  from 
April  12  to  Aug.  8  ?  to  Aug.  3  ? 

2.  What  change  would  be  made  in  the  number  of  days 
given  in  the  table  if  Feb.  29  of  a  leap  year  should  intervene 
between  dates  ? 

332.  Written  Exercises. 

Using  the  table  in  Sec.  331,  find  by  the  60-day  method 
(p.  212)  the  interest  on : 

1.  $  2000  from  April  1  to  July  4  at  6  %. 

2.  $  4500  from  Dec.  20  to  Feb.  15  at  7  %. 

3.  $  20,000  from  May  12  to  Aug.  5  at  5  %. 

4.  $  80,000  from  April  16  to  June  4  at  6  %. 

5.  $25,000  from  July  4  to  Sept.  16  at  8  %. 

6.  $4370  from  Aug.  4  to  Dec.  7  at  6  %. 

EXACT   INTEREST 

333.  1.  In  the  methods  of  computing  interest  in  Sees.  255- 
261,  360  days  were  taken  as  one  year  in  reckoning  the  interest 
for  periods  less  than  a  year.  Interest  computed  on  the  basis 
of  365  days  to  the  year  is  called  exact  interest. 

2.  Exact  interest  is  found  by  taking  such  a  fractional  part 
of  a  year's  interest  as  the  exact  number  of  days  is  of  365  days. 

3.  Find  the  exact  interest  on  $  400  from  March  2  to  June 

15  at  8%. 

Find  the  exact  number  of  days  from  March  2  to  June  15. 
Do  not  count  March  2,  but  count  June  15.  From  March  2  to 
June  15  there  are  29  da.  +  30  da.  +  31  da.  +  15  da.  or  105  da. 

Model  :  21 

$400  X  .08  X  W  =  e9.20+. 

m 

78 


TAXES  26d 

4.  Compare  the  interest  on  $  400  for  105  da.  at  8  %  com- 
puted in  the  ordinary  way  with  the  exact  interest. 

5.  Exact  interest  for  whole  years  is  the  same  as  interest 
computed  by  the  ordinary  method.  For  periods  of  less  than  a 
year,  the  exact  interest  is  |f-|  of  the  interest  computed  by  the 
ordinary  method,  or  -^-^  (y^g)  less  than  the  ordinary  interest. 
Compared  with  interest  computed  by  the  ordinary  method,  is 
exact  interest  favorable  to  the  person  loaning  money  or  to  the 
person  borrowing  it  ? 

6.  Exact  interest  is  computed  by  the  United  States  govern- 
ment and  sometimes  by  others. 

7.  How  much  interest  does  the  United  States  government 
save  by  paying  exact  interest  rather  than  ordinary  interest  on 
$  5,000,000  from  Jan.  1  to  April  1, 1907,  at  3  %  ? 

8.  Find  exact  interest  on  each  of  the  amounts  in  Sec.  332 
for  the  time  specified. 

TAXES 

334.  1.  Every  person  in  the  United  States  derives  some 
benefits  from  one  or  more  of  the  following  divisions  of  govern- 
ment :  national  (also  called  federal),  state,  county,  township,  school 
district,  city,  and  village.  Explain  the  purpose  of  each  of  these 
divisions  of  government  and  name  some  of  the  benefits  you 
derive  from  each.  The  maintenance  of  these  several  forms 
of  government  necessarily  involves  the  expenditure  of  money. 
This  money  is  derived  from  taxes  levied  upon  the  persons, 
property,  incomes,  or  business  of  individuals.  Mention  some 
purpose  for  which  money  is  expended  by  each  division  of 
government  named  above. 

2.  Direct  taxes  are  sums  of  money  levied  upon  persons,  prop- 
erty, incomes,  or  business  of  individuals  for  the  support  of 
state  or  local  governments.  They  are  called  direct  taxes  be- 
cause they  are  levied  directly  upon  the  persons,  lands,  build- 
ings, etc.,  of  individuals  and  are  payable  directly  to  public 
officials  authorized  to  collect  taxes. 


270  APPENDIX 

3.  Indirect  taxes  are  sums  levied  upon  goods  imported  from 
other  countries,  upon  certain  products  manufactured  in  our 
own  country,  and  upon  the  privilege  of  engaging  in  certain 
pursuits,  as  selling  liquors,  etc.  They  are  called  indirect  taxes 
because  they  are  paid  indirectly  by  the  consumers. 

4.  Ascertain  how  the  money  is  raised  for  building  school- 
houses,  for  repairing  roads  or  streets,  etc.,  in  the  community 
in  which  you  live. 

STATE   AND   LOCAL   TAXES 

335.  Poll  Tax.  Each  male  inhabitant  of  voting  age  (generally 
with  certain  exceptions)  is  usually  taxed  a  fixed  sum  annually, 
regardless  of  the  ownership  of  taxable  property.  This  is  called  a 
poll  tax.  Poll  means  head.  Is  a  poll  tax  a  direct  or  an  indirect 
tax  ?  It  is  the  only  form  of  personal  tax  levied  in  the  United 
States.  What  are  the  provisions  relating  to  poll  tax  in  your  state  ? 

336.  1.  Property  Tax.  Property  is  classified  either  as  per- 
sonal property  or  as  real  property.  All  movable  property,  as 
household  goods,  money,  cattle,  ships,  etc.,  is  called  personal 
property.  Immovable  property,  as  lands,  buildings,  mines,  etc., 
is  called  real  property.  Both  forms  of  property  are  subject  to 
taxation. 

2.  For  the  purpose  of  taxation,  the  value  of  all  taxable  prop- 
erty is  estimated  by  a  public  officer  called  an  assessor.  This 
officer  prepares  a  list  of  all  taxable  property  in  his  district, 
showing  the  names  of  the  owners,  the  location  of  the  property, 
and  its  assessed  valuation.     Such  a  list  is  commonly  called  an 

.assessment  roll.     The  assessed  valuation  of  property  is  gen- 
erally less  than  its  actual  value. 

3.  Taxes  are  usually  collected  by  a  public  officer  called  a 
tax  collector,  who  deposits  the  amounts  collected  with  another 
public  officer  called  a  treasurer. 

4.  What  is  public  property  ?  What  properties,  other  than 
public  property,  are  exempt  from  taxation  in  your  state  ? 


CUSTOMS  S71 

5.  The  total  assessed  valuation  of  the  property  in  any  state 
is  the  sum  of  the  assessed  valuation  of  the  property  in  the 
several  counties  of  the  state.  How  is  the  assessed  valua- 
tion of  the  county  in  which  you  live  determined  ?  The  state 
and  local  taxes  are  usually  levied  and  collected  together.  As- 
certain how  state,  county,  and  local  taxes  are  levied  and  col- 
lected in  your  state. 

6.  Examine  a  receipt  for  the  payment  of  taxes  on  real 
property.     How  is  the  location  of  the  property  described? 

337.  Written  Exercises. 

1.  The  assessed  valuation  of  a  certain  county  is  $8,000,000. 
Find  the  rate  of  taxation  that  will  yield  revenue  sufficient  to 
pay  for  a  new  courthouse  costing  $  100,000.  How  much  must 
Mr.  Thomas  pay  toward  the  building  of  the  courthouse  if  his 
taxable  property  is  assessed  at  $4000? 

2.  Find  the  value  of  Mr.  White's  property  if  his  taxes 
amount  to  $67.50  when  the  tax  rate  is  $.01|-  and  his  property 
is  assessed  at  |  of  its  value. 

3.  The  rate  of  taxation  in  a  certain  county  is  $.01|  when 
the  assessed  valuation  is  f  of  the  actual  value  of  the  property. 
What  would  the  rate  of  taxation  have  been  to  yield  the  same 
amount  had  the  property  been  assessed  at  its  full  value  ? 

4.  Make  and  solve  five  problems  in  taxes,  using  if  possible 
the  actual  tax  rates  of  the  community  in  which  you  live. 

CUSTOMS  AND  INTERNAL  REVENUE 

338.  1.  Funds  for  the  maintenance  of  the  national  gov- 
ernment are  derived  chiefly  from  customs  and  internal  revenue. 

2.  Customs,  or  duties,  are  taxes  levied  by  the  national  govern- 
ment upon  goods  imported  from  other  countries.  Internal 
revenue  is  a  tax  levied  by  the  national  government  upon  the 
manufacture  or  sale  of  certain  articles  in  the  United  States. 


272  APPENDIX 

339.  The  following  tables  show  the  receipts  and  expendi- 
tures of  the  national  government  during  the  fiscal  year  ended 
June  30,  1906: 

Revenues,  Fiscal  Year  190G 

Customs $300,251,877.77 

Internal  Revenue 249,150,212.91 

Lands 4,879,833.05 

Miscellaneous 40,172,197.34 

Total,  exclusive  of  Postal       ....  ^594,464,121.67 

Expenditures,  Fiscal  Year  1906 

Civil  and  Miscellaneous $159,823,904.50 

War  Department 119,704,113.09 

Navy  Department 111,166,784.35 

Indians 12,746,859.08 

Pensions 141,034,561.77 

Interest 24,308,576.27 

Total,  exclusive  of  Postal       ....       $508,784,799.06 

The  following  table  shows  the  receipts  from  the  principal 
objects  of  internal  taxation  during  the  same  fiscal  year : 

Internal  Revenue  Receipts,  Fiscal  Year  1906 

Distilled  Spirits .        .  $  143,394,055.00 

Tobacco 48,422,997.88 

Fermented  Liquors 55,641,858.56 

Oleomargarine 570,037.93 

Mixed  Flour 2,567.23 

Adulterated  Butter 9,258.43 

Renovated  Butter 138,078.09 

Playing  Cards 489,347.26 

Penalties 283,901.62 

Collections 150,494.88 

340.  1.  Customs.  A  list  or  schedule  of  goods  with  the  rates 
of  import  duties  adopted  by  Congress  is  called  a  tariff.  Under 
our  tariff  laws  some  imported  articles  are  admitted  without  the 
payment  of  duties.  These  articles  are  said  to  be  on  the  free 
list.  Articles  not  on  the  free  list  are  subject  to  an  ad  valorem 
duty,  a  specific  duty,  or  to  both. 


CUSTOMS  273 

2.  An  ad  valorem  duty  is  a  duty  reckoned  according  to  the 
value  or  cost  of  the  goods  in  tlie  country  from  whicli  they  are 
imported.     Thus,  the  duty  on  jewelry  is  60  %  of  the  value. 

3.  A  specific  duty  is  a  tax  of  a  specified  amount  on  each 
pound,  yard,  gallon,  bushel,  etc.,  regardless  of  the  cost  of  the 
goods.     Thus,  the  duty  on  onions  is  40)^  a  bushel. 

4.  Certain  ports  are  designated  sls  ports  of  entry,  where  duties 
on  cargoes  are  payable.  A  customhouse  has  been  established 
at  each  port  of  entry  for  the  collection  of  customs.  The  col- 
lection of  customs  at  each  port  is  under  the  direction  of  a 
government  officer  called  the  collector  of  the  port. 

5.  The  following  rates  of  custom  are  from  the  schedule 
adopted  by  Congress  in  1897,  and  commonly  known  as  the 
Dingley  Tariff. 

Cheese,  6^  per  lb.  Paintings,  20  %  ad  val. 

Hay,  $4:  per  T.  Penholders,  25  %  ad  val. 

Coffee,  free.  Carpets  (Axminster),  60/  per 
Apples,  Dried,  2/  per  lb.  sq.  yd.  and  40  %  ad  val. 

Tea,  free.  Carpets  (Brussels),  44/  per  sq. 
Bacon  and  Hams,  5/  per  lb.  yd.  and  40% ad  val. 

Honey,  20  /  per  gal.  Oats,  15/ per  bu. 

Soap  (Castile),  Ij/  per  lb.  Wheat,  25/  per  bu. 

Musical  instr'ts,  45%  ad  val.  Hops,  12/  per  lb. 

341.   Written  Exercises. 

1.  How  much  is  the  duty  on  a  violin  worth  $  80  ? 

2.  A  painting  costing  $2500  was  purchased  in  Italy  and 
brought  into  the  United  States.  Find  the  amount  of  the  duty 
charged  for  its  admission. 

3.  What  is  the  duty  on  200  sq.  yd.  of  Brussels  carpet  valued 
at  $1.20  a  square  yard  ? 

4.  What  per  cent  of  the  expenditures  of  the  national  gov- 
ernment were  defrayed  from  customs  receipts  in  1906  ?  from 
internal  revenue  receipts  ? 

AR.— 18 


274 


APPENDIX 


BANKING 

342.  Give  the  names  of  some  banks  that  you  know  of.     Of 

what  use  to  a  community  are  banks  ?  Why  is  a  bank  a  safer 
place  in  which  to  keep  money  than  a  house  or  an  office  ?  If  a 
person  wishes  to  make  a  payment  to  another,  he  may  do  so  by 
giving  him  the  money.  Do  you  know  of  any  other  method 
commonly  used  in  making  payments  ? 

343.  Savings  Banks.  Savings  banks  are  banks  organized 
under  the  laws  of  the  different  states  for  the  purpose  of  re- 
ceiving and  investing  the  savings  of  people.  Their  capital 
consists  of  the  money  put  into  the  bank  by  the  depositors,  and 

Deposit  Slip.  their     profits     are     divided 

among  the  depositors  in  pro- 
portion to  the  amount  that 
each  has  on  deposit.  The 
profits  are  paid  to  the  depos- 
itors in  the  form  of  interest, 
usually  ranging  from  3%  to 
4%  annually,  which  is  paid 
monthly,  quarterly,  or  semi- 
annually. If  the  interest  is 
not  collected  by  the  depositor 
when  it  becomes  due,  it  is 
entered  to  his  credit  on  the 
books  of  the  bank  and  it 
thereafter  draws  interest  in 
the  same  manner  as  the 
ordinary  deposits.  Do  sav- 
ings banks  pay  compound 
interest  ? 

344.  Savings  Bank  Ac- 
counts. Any  reliable  person 
who  wishes  to  deposit  money 
for  safekeeping  and  invest- 
ment may  open  an   account 


SAVINGS  DEPOSIT 

No.  f^8S         Bal.  f  3/6.^6 
deposited  with 

Union  Savings  Bank 

FOR  ACCOUNT  OF 
Lo8  Angklbs,  Cal.,  March  22, 1907. 

Gold  .... 
Silver    .     .     . 

Currency  .    . 

Checks  .    .    . 

<< 

« 

DOLLARS 

10 

BANKING  275 

with  a  savings  bank.  On  opening  an  account,  the  depositor 
is  usually  required  to  answer  certain  questions  and  to  leave  his 
signature  with  the  bank,  to  protect  the  bank  against  fraudulent 
demands  upon  his  accounts.  The  depositor  then  hands  the 
"receiving  teller"  or  the  "cashier"  the  money  which  he 
wishes  to  deposit,  together  with  a  deposit  slip,  showing  the 
amount  of  his  deposit.  He  is  then  given  a  savings  bank  book, 
usually  bearing  a  number,  with  the  amount  of  his  deposit 
credited  to  his  account.  This  bank  book  must  be  presented 
whenever  the  depositor  wishes  to  make  a  deposit  or  to  draw 
against  his  credit  in  the  bank.  The  smallest  amount  received 
for  deposit  is  usually  one  dollar. 

When  a  depositor  withdraws  money  from  a  savings  bank,  he  is 
required  to  give  the  bank  a  receipt  for  the  amount  he  has 
withdrawn. 

Receipt 


SAVINGS  ACCOUNT 
Los  Angeles,  Cal.,  {Z^ll  f6,  /(^OJ       No.  1^88 
Received  from  the  UNION   SAVINGS   BANK 

c^(:/^^K.  o^^  f^-....^.^..-..-^.--^.--^  f  15.00 

Balance,  f^85./S.    . 


As  the  usage  of  banks  differs  considerably  in  the  various 
sections  of  the  country,  no  attempt  is  made  here  to  give 
details.  Pupils  should  familiarize  themselves  with  local 
customs.  Deposit  slips,  receipt  blanks,  or  check  blanks,  sam- 
ple bank  books,  note  forms,  etc.,  should  be  examined  by  the 
pupils  and  their  use  explained  to  them.  Where  possible,  a 
visit  to  a  neighboring  bank  should  be  made  during  banking 
hours. 


276 


APPENDIX 


345.    Illustrative  page  from  a  savings  bank  book,  showing 
entries  of  deposits,  withdrawals,  and  interest: 


Statement 


Date 

Withdrawn 

DEPOsrrED 

Balance 

Dec.  6,  1907  .... 

45 

50 

128 

75 

Dec.  21,  1907  .  .  . 

15 

143 

75 

Dec.  24,  1907  .  .  . 

35 

108 

75 

Int.  to  Jan.  1,  1908  . 

4 

74 

113 

49 

Jan.  3,  1908  .... 

50 

163 

49 

Feb.  2,  1908  .... 

74 

50 

237 

99 

March  7,  1908  .  .  . 

100 

137 

99 

March  25,  1908  .  .  . 

80 

217 

99 

346.  Banks  of  Deposit.  Banks  other  than  savings  banks  are 
sometimes  called  haiiks  of  deposit.  They  are  variously  known 
as  national  banks,  state  banks,  commercial  banks,  private  banks, 
etc.  These  banks  are  organized  for  the  purpose  of  receiving 
deposits,  making  loans,  etc.  National  banks  are  organized 
under  the  national  law  and  are  under  the  direct  supervision  of 
the  Comptroller  of  Currency,  who  is  appointed  by  the  President 
of  the  United  States.  In  addition  to  carrying  on  a  general 
banking  business,  national  banks  have  authority  to  issue  paper 
money,  called  bank  notes.  The  payment  of  these  notes  is 
secured  by  government  bonds  deposited  with  the  Secretary  of 
the  Treasury.  All  other  banks  are  organized  under  the  laws 
of  the  states  in  which  they  are  located. 

347.  Deposits.  Accounts  are  opened  with  banks  of  deposit 
in  much  the  same  manner  as  with  savings  banks.  Business 
men  and  others  who. wish  to  keep  money  on  hand  for  the  pay- 
ment of  bills,  etc.,  usually  have  an  account  with  a  bank  against 
which  they  may  draw  checks  whenever  they  wish.  Such  ac- 
counts usually  do  not  draw  interest.  Each  depositor  receives 
a  hank  book  in  which  all  deposits  must  be  entered. 


BANKING 


277 


348.  1.  Checks.  When  the  account  is  opened,  the  depositor 
receives  a  check  hook  which  he  nses  in  making  demands  against 
his  accoimts. 


Stub 


Check 


San  Francisco,  Cal.,  ?]1a.if  ^',  /^OS       No.  68 

UNION   NATIONAL   BANK 
Pay  to  the  order  of 

<S^kvityY-iiV90  am^cL  ■p-N.^N^N^-s.r>^'-v/-%^DoLLAKS 


2.  Checks  are  indorsed  in  the  same  manner  as  promissory- 
notes. 

3.  To  cash  this  check,  James  E.  Thomas  will  indorse  it  and 
present  it  at  the  bank,  or  will  deposit  it  to  his  credit  at  the 
bank  where  his  accounts  are  kept.  The  check  will  finally  be 
returned  to  R.  E.  Davies  after  it  has  been  paid.  It  will  then 
serve  as  a  receipt  from  James  E.  Thomas,  since  it  bears  his 
indorsement. 

4.  A  depositor  may  draw  money  for  himself  from  his  bank 
by  making  his  check  payable  to  "  Cash,"  in  which  case  no  in- 
dorsement is  necessary. 

349.  Certificate  of  Deposit.  When  a  person  who  does  not 
intend  to  become  a  regular  customer  of  a  bank  makes  a  deposit, 
he  is  given  a  certificate  of  deposit,  showing  the  amount  of  his 
deposit.  Such  a  deposit  is  not  subject  to  check.  The  amount 
may  be  withdrawn  upon  the  return  of  the  certificate  with  the 
proper  indorsement.  Certificates  of  deposit  are  usually  issued 
to  persons  who  deposit  money  with  banks  when  interest  is  paid 
upon  the  deposit. 


278  APPENDIX 

350.  Drafts.  Banks  usually  keep  money  on  deposit  in 
some  bank,  called  a  correspondence  bank,  in  Boston,  New  York, 
Chicago,  San  Francisco,  or  other  financial  center,  against  which 
to  draw  checks.  When  a  person  wishes  to  make  a  payment  in 
a  distant  place,  he  may  purchase  the  check  of  his  local  bank  on 
a  bank  of  correspondence,  which  will  honor  this  check  when 
presented  for  payment.  Such  a  check  is  called  a  bank  draft. 
Thus,  Mr.  A,  in  Seattle,  Wash.,  may  wish  to  pay  Mr.  B,  in 
Peoria,  111.,  $45.60.  Mr.  A  may  purchase  a  draft  of  a  bank 
in  Seattle  on  a  bank  in  Chicago,  which  will  honor  the  draft 
when  presented  either  by  Mr.  B,  or  by  some  bank  which  has 
purchased  it  from  Mr.  B.  The  bank  in  Seattle  may  charge 
Mr.  A  a  small  amount  for  making  this  exchange  of  money. 

351.  Clearing  House.  Daily  settlements  of  accounts  between 
banks  are  made  through  an  association  called  a  clearing  house. 
At  a  fixed  hour  each  day  representatives  from  each  bank  that 
is  a  member  of  the  clearing  house  visit  the  clearing  house 
and  settle  the  accounts  of  their  bank  with  other  banks. 
All  large  cities  have  clearing  houses  and  nearly  all  banks  in 
these  cities  are  members. 

By  means  of  the  clearing  house  the  American  Exchange  National 
Bank  one  day  transacted  a  business  of  $  18,000,000  in  checks,  with  a  bal- 
ance of  only  12  cents  to  pay.  The  clearing  house  sheets  showed  that 
$9,049,255.40  in  checks,  drawn  by  the  depositors  of  the  bank,  had  been 
turned  in  by  other  institutions.  Against  these  the  bank  had  .$  9,049,255.28 
in  checks  of  other  banks  belonging  to  the  clearing  house,  which  had  been 
deposited  with  the  American  Exchange  National  Bank.  The  clearance 
was  made  by  a  payment  by  the  bank  of  12  cents  to  the  clearing  house. 

LIFE    INSURANCE 

352.  Personal  Insurance.  There  are  various  kinds  of  per- 
sonal insurance.  Of  these  the  most  common  are:  Accide^it 
insurance,  which  is  an  indemnity  for  injuries  sustained  by 
accident ;  health  insurance,  which  is  an  indemnity  for  loss 
of  time  caused  by  illness ;  life  insurance,  the  principal  forms 
of  which  are  discussed  on  p.  279. 


LIFE  INSURANCE  279 

353.  1.  Persons  who  insure  their  lives  usually  do  so  to  pro- 
vide for  those  who  are  dependent  upon  them.  The  cost  of  life 
insurance  depends  (a)  upon  the  age  of  the  person  insured; 
and  (b)  upon  the  kind  of  policy  taken  out.  Life  insurance 
premiums  are  always  stated  at  so  much  on  each  $1000  of 
insurance.     The  most  common  kinds  of  policies  are  : 

2.  Ordinary  Life  Policies,  called  also  straight  life  policies  and 
life  policies.  The  insured  pays  a  premium,  usually  annually, 
at  the  beginning  of  each  year  from  the  time  he  insures  his  life 
until  his  death.  At  his  death,  the  company  pays  the  face  of 
the  policy  to  the  person  (or  persons)  named  in  the  policy  as 
his  beneficiary. 

3.  Limited  Payment  Life  Policies.  The  insured  pays  a  pre- 
mium for  a  limited  number  of  years,  as  20  years,  at  the  expira- 
tion of  which  the  policy  is  said  to  be  paid  tip.  The  face  of  the 
policy  is  paid  to  the  beneficiary  at  the  death  of  the  insured. 

4.  Endowment  Policies.  Premiums  are  paid  for  a  period  of 
years,  as  10,  15,  or  20  years,  and  the  face  of  the  policy  is  paid 
at  the  end  of  the  period  specified,  or  at  death  if  the  insured 
should  die  before  the  expiration  of  the  period. 

5.  Term  Policies.  The  insurance  extends  for  a  specified 
period,  as  for  10,  15,  20  years,  etc.,  at  the  expiration  of  which 
the  insurance  ceases.  The  face  of  the  policy  is  paid  if  the 
insured  dies  within  the  period  specified. 

The  amount  of  the  annual  premium  on  $1000  of  insurance  for  a  life 
policy,  a  limited  life  policy,  and  for  an  endowment  policy,  ages  20  years 
to  40  years,  is  given  in  the  table  on  j).  280. 

6.  Which  is  the  more  likely  to  live  twenty  years  longer,  a 
person  twenty  years  of  age  or  a  person  forty  years  of  age? 
Statistics  have  been  carefully  compiled  showing  the  ages  at 
which  persons  die.  From  these  statistics  insurance  companies 
are  able  to  determine  the  average  number  of  years  a  healthy 
person  of  a  given  age  may  be  expected  to  live.  The  rates  of 
annual  premiums  are  based  upon  these  statistics. 


280 


APPENDIX 


Tablb  or  Annual  Premiums  for  $  1000  (Ages  20  years  to  40  years) 

Policies  Non-fobfbitable  and  Participatinq 

Premiums  may  also  be  paid  half-yearly  or  quarterly ;  and  if  desired,  may  be 
paid  iu  10,  15,  or  20  years  instead  of  during  the  whole  term. 


Ordinary  Life 

20 

Endowments 

J 

Payment 
Life 

Age 

Yearly 

10  Year 

15  Year 

20  Year 

20 

$17.30 

$24.16 

$99.27 

$62.34 

$44.10 

21 

17.80 

24.00 

99.40 

62.40 

44.25 

22 

18.30 

25.10 

99.50 

62.45 

44.40 

23 

18.70 

25.70 

99.00 

62.50 

44.55 

24 

19.30 

26.20 

99.75 

62.G0 

44.70 

25 

19.80 

26.75 

99.90 

62.70 

44.82 

26 

20.30 

27.30 

100.00 

62.80 

44.95 

27 

20.90 

27.90 

100.05 

62.90 

45.10 

28 

21.50 

28.50 

100.10 

63.05 

45.25 

29 

22.10 

29.10 

100.20 

63.20 

45.45 

30 

22.70 

29.70 

100.30 

63.34 

45.63 

31 

23.40 

30.35 

100.40 

63.i30 

45.85 

32 

24.10 

31.00 

100.50 

63.70 

46.05 

33 

24.80 

31.72 

100.60 

63.90 

46.25 

34 

25.60 

32.50 

100.75 

64.05 

46.45 

35 

26.50 

33.28 

100.90 

64.20 

46.70 

36 

27.40 

34.10 

101.15 

64.40 

46.85 

37 

28.30 

34.96 

101.45 

64.65 

47.05 

38 

29.30 

35.88 

101.75 

64.95 

47.25 

39 

30.40 

36.84 

101.95 

65.30 

47.45 

40 

31.50     j 

37.84 

102.14 

65.67 

48.64 

364.  Dividends.  The  premium  charged  represents  the  esti- 
mated cost  of  insurance  and  is  based  upon  conservative  assump- 
tions as  to  future  death  rate,  the  rate  of  interest  which  the 
company  may  expect  to  receive  for  loans,  etc.  The  actual  cost 
of  insurance  is  determined  by  experience  from  year  to  year. 
The  difference  between  the  estimated  cost  and  the  actual  cost 
J 8  called  the  profit.     Policy  holders  are  usually  allowed  to 


LIFE  INSURANCE  281 

participate  in  the  profits,  either  by  having  them  applied  to 
reduce  the  yearly  premiums  or  by  having  them  accumulate  in 
the  possession  of  the  companies  until  the  expiration  of  the 
term  of  insurance.  An  insurance  policy  in  which  it  is  stipu- 
lated that  no  dividend  shall  be  paid  until  the  close  of  the  term 
of  insurance  is  called  a  tontine  policy. 

Examine  a  life  insurance  policy.     Kead  all  its  provisions. 

355.    Use  the  table  in  answering  the  following : 

1.  How  much  will  it  cost  annually  to  carry  an  ordinary  life 
policy  for  $  1000,  if  it  is  taken  out  at  the  age  of  20  ?  at  the 
age  of  25  ?  at  the  age  of  35  ? 

2.  How  much  will  it  cost  annually  to  carry  a  20-payment 
life  policy  for  $2000,  if  it  is  taken  out  at  the  age  of  20?  at 
the  age  of  27  ?  at  the  age  of  40  ? 

3.  How  much  will  it  cost  annually  to  carry  a  10-year  en- 
dowment policy  for  $  5000,  if  it  is  taken  out  at  the  age  of  20  ? 
at  the  age  of  30  ?  at  the  age  of  40  ? 

4.  Suppose  that  a  young  man  20  years  old  takes  out  a  20- 
payment  life  policy  for  f  1000  and  dies  after  paying  8  annual 
premiums.  Find  the  net  cost  of  the  insurance,  if  dividends 
amounting  to  $40  were  applied  to  reduce  the  premiums.  How 
much  would  the  beneficiary  named  in  the  policy  receive  at  his 
death  ? 

5.  How  much  will  it  cost  to  carry  a  20-year  endowment 
policy  for  $  1000  for  the  term  of  the  policy,  if  it  is  taken  out 
at  the  age  of  30  ?  How  much  would  the  insured  receive  from 
the  insurance  company  at  the  end  of  the  term,  not  including 
the  dividends  ? 

6.  If  the  insured  (Prob.  5)  died  after  paying  15  premiums, 
how  much  more  than  the  amount  paid  as  premiums  would 
the  beneficiary  receive  ? 

7.  What  is  meant  by  a  non-forf citable  and  participating 
policy  ?  by  a  tontine  policy  ? 


282 


APPENDIX 


Table  of  Loak  and  Surrender  Values 

356.  The  following  table  shows  the  loan  and  surrender 
values  on  a  20-payment  life  policy  for  $  1000  taken  out  when  the 
insured  was  25  years  of  age,  the  annual  premium  being  $  26.95 : 


At  End  of 

Loan 

Cash 
Value 

Pald-up 
Insurance 

Extended 
Insurance 

Years          Days 

3d 

«150 

4            342 

4th 

200 

6            291 

5th 

$54 

$60 

250 

8            232 

6th 

68 

76 

300 

10            317 

10th 

130 

145 

500 

19              17 

15th 

224 

249 

750 

26            134 

19th 

315 

351 

950 

31            111 

20th 

342 

380 

Policy  full-paid 

Answer  the  following  from  the  above  table : 

1.  If  the  insured  wished  to  borrow  money,  how  much  would 
the  company  loan  him  at  the  end  of  the  10th  year,  if  he 
assigned  to  the  company  his  policy  as  security  ?  how  much  at 
the  end  of  the  15th  year  ? 

2.  If  the  insured  surrendered  his  policy  at  the  end  of  the 
5th  year,  how  much  would  the  insurance  company  pay  him 
for  his  policy  ?  How  long  would  they  continue  his  insurance 
without  the  payment  of  premiums  ? 

3.  Find  the  amount  of  the  annual  premiums  for  20  years. 
What  is  the  cash  value  of  the  policy  at  the  end  of  20  years  ? 
If  the  dividends  average  $  6.50  a  year,  how  much  will  they 
amount  to  in  20  years  ?  What  is  the  sum  of  the  cash  value 
and  dividends  at  the  end  of  the  insurance  term  ?  How  does 
this  sum  compare  with  the  total  cost  of  the  insurance  for  the 
term  ? 


THE  EQUATION  283 

4.  Using  the  compound  interest  table  on  p.  320,  find  the 
amount  of  $  26.95  (the  premium)  for  20  years.  If  money  is  worth" 
6%,  find  the  total  amount  of  the  premiums  paid  at  the  end  of 
20  years,  the  premium  being  paid  at  the  beginning  of  each  year. 

5.  If  the  insured  should  die  at  the  age  of  40,  how  much 
would  the  beneficiary  receive  ? 

THE   EQUATION 

357.  1.  The  relation  of  the  quantities  involved  in  some  prob- 
lems can  be  stated  in  a  simpler  and  clearer  way  by  the  use 
of  the  equation.  In  an  equation,  the  vahie  of  the  unknown 
quantity  is  usually  represented  by  the  letter  x.  Thus,  in 
6  -f  4  =  0^,  X  is  called  the  unknown  quantity,  and  the  expres- 
sion 6  -h  4  =  ^'  is  called  an  equation. 

2.  An  equation  may  be  compared  to  a  balance  scale.  In 
an  equation  the  quantities  on  the  two  sides  are  equivalent  — 
they  balance  one  another. 

3.  If  a  package  weigh- 
ing 4  lb.  is  placed  in  one 
pan  of  a  balance  scale, 
what  weight  must  be 
placed  in  the  other  pan  to 
make  the  scale  balance  ? 

4.  A  package  weighing  5  lb.  was  placed  in  the  pan  on  the 
right  of  a  balance  scale  and  a  2-lb.  weight  was  placed  in  the 
pan  on  the  left.  What  additional  weight  must  be  placed  in 
the  pan  on  the  left  to  make  the  scales  balance  ? 

5.  Would  the  scales  as  represented  in  the  figure  still  bal- 
a!ice  if  a  5-lb.  weight  were  added  to  the  weights  in  each  pan  ? 
Would  they  balance  if  a  5-lb.  weight  were  added  to  the  weight 
in  one  pan  ? 

6.  Would  the  scales  as  represented  in  the  figure  still  bal- 
ance if  2  lb.  were  removed  from  both  pans  ?  Would  they 
balance  if  2  lb.  were  removed  from  only  one  pan  ? 


284  APPENDIX 

7.  Would  the  scales  as  represented  in  the  figure  balance  if 
the  weight  on  both  sides  were  doubled  ?  Would  they  balance 
if  the  weight  on  only  one  side  were  doubled  ? 

8.  Would  the  scales  as  represented  in  the  figure  balance  if 
one  half  of  the  present  weight  were  removed  from  each  pan  ? 
Would  they  balance  if  one  half  of  the  weight  were  taken  out 
of  only  one  pan  ? 

9.  What  is  the  value  of  the  unknown  quantity  in  the 
equations  lb.  =5  lb.  +  a;?  in  6  lb. +  a;  =  10  lb.  ?  inZlb.  — «  = 
2  1b.? 

10.  What  is  the  value  of  x  in  9  +  a?  =  12  ?  Finding  the 
value  of  the  unknown  quantity  in  an  equation  is  called  solving 
the  equation. 

11.  If  3  is  added  to  both  sides  of  the  equation  a;  +  4  =  0, 
the  result  is  a;  +  7  =  12.  How  does  the  value  of  a;  in  a;  +  4  =  9 
compare  with  the  value  ofa;ina;-l-7  =  12? 

12.  Write  5  equations.  Add  some  number  to  both  sides  of 
each  of  the  equations.  Compare  the  value  of  x  in  the  result- 
ing equation  with  the  value  of  x  in  the  original  equation. 

13.  State  what  effect  adding  the  same  number  to  both  sides 
of  the  equation  has  upon  the  value  of  x  in  the  equation. 
Prove  the  truth  of  your  statement. 

14.  If  2  is  subtracted  from  both  sides  of  the  equation 
a;  -I-  4  =  9,  the  result  is  a;  +  2  =  7.  How  does  the  value  of  x 
in  a;  -f-  2  =  7  compare  with  the  value  of  a;  in  a;  +  4  =  9  ? 

15.  Write  5  equations.  Subtract  some  number  from  both 
sides  of  each  equation.  Compare  the  value  of  x  in  each  of  the 
resulting  equations  with  the  value  of  x  in  the  original  equation. 

16.  State  what  effect  upon  the  value  of  x  in  any  equation 
subtracting  the  same  number  from  both  sides  of  the  equation 
has  upon  the  equation.     Prove  the  truth  of  your  statement. 

17.  If  4  is  subtracted  from  both  sides  of  the  equation 
jc  -f  4  =  9,  what  is  the  result  ? 


THE  EQUATION         .  285 

18.  If  3  is  subtracted  from  both  sides  of  the  equation  x  -\-  S 
—  8,  the  result  is  a?  =  5.  If  2  is  subtracted  from  both  sides  of 
the  equation  9  =  2  +  a;,  the  result  is  7  =x.  What  number 
must  be  subtracted  from  both  sides  of  each  of  the  following 
equations  to  leave  x  bj  itself  on  one  side:  a? 4- 5  =  8? 
a;  +  7  =  15?  x  +  6  =  9?  S  =  6-\-x?  12  =  x  +  o?  9-f-ic=15? 
74-aJ  =  10? 

19.  If  4  is  added  to  both  sides  of  the  equation  a;  —  4  =  5, 
the  result  is  a;  =  9. 

20.  What  number  must  be  added  to  both  sides  of  each  of 
the  following  equations  to  leave  x  by  itself  on  one  side : 

iB-5  =  8?     a;-7  =  10?     a;-4  =  ll?     x-0  =  2? 
12  =  a;-15?     10  =  a;-5?        4  =  a;-G?     2  =  a;-3? 

21.  The  value  of  a;  in  a;  +  4  =  9  may  be  found  by  subtract- 
ing 4  from  the  left  side  of  the  equation  and  indicating  the  sub- 
traction of  4  from  the  other  side,  thus :  a;  =  9  —  4.  Find  the 
value  of  X  in  each  of  the  following  equations  :  a;  -f-  6  =  15 ; 
fl;  +  8  =  15;     a;  +  9  =  16;     a;-f-20  =  45;      a;-f345  =  670. 

22.  The  value  of  a;  in  a;  —  4  =  10  may  be  found  by  adding  4 
to  the  left  side  of  the  equation  and  indicating  the  addition  of  4 
to  the  other  side,  thus :  a;  =  10  +  4.  Find  the  value  of  x  in 
each  of  the  following  equations  :  a;  —  7  =  13  ;  a;  —  5  =18  ; 
a;-12  =  20;    a;-14  =  17;   a;-46  =  35;    a;-80  =  120. 

23.  Write  each  of  the  following  equations  with  x  by  itself  on 
the  left  side  of  the  equation :  a?  +  3  lb.  =  10  lb. ;  5  f t.  +  a'  =  13  ft. ; 
24  yd. +a;  =  45yd.;  $  7.50 +  aj=  $12.75;  a; +  $15  =  $80; 
a; -12  ft.  =20  ft.;     a; -  $3.45  =  $1.20. 

24.  Compare  2  +  3  =  5  with  5  =  2  +  3.  Compare  a;  +  4  =  9 
with  9  =  a;  +  4.  State  what  effect,  if  any,  writing  the  equation 
with  the  sides  changed  has  upon  the  equation. 

25.  Write  each  of  the  following  equations  so  the  side  con- 
taining X  is  on  the  left :  45  ft.  +  33  f t.  =  a; ;  $  2.45  =  a;  -  $  1.20 ; 
14  yr.  =  9  yr.  +  a; ;  10  yr.  =  a;  —  7  yr. 


286  APPENDIX 

358.  Solve  each  of  the  following  without  using  x.  Then 
write  the  equation  for  each,  using  x^  and  find  the  value  of  x : 

1.  If  45  is  added  to  a  certain  number,  the  sum  is  73.  What 
is  the  number  ? 

Model  :    Let  x  =  the  unknown  number 
a;  +  45  =  73 

a  =  73  -  46 
a;  =  28 

2.  If  27  is  subtracted  from  a  certain  number,  the  remain- 
der is  56.     What  is  the  number  ? 

3.  If  a  certain  number  is  increased  by  347,  the  result  is 
591.     What  is  the  number  ? 

4.  If  a  certain  number  is  diminished  by  274,  the  result  is 
483.     What  is  the  number  ? 

5.  A  boy  deposited  ^  17  in  a  savings  bank.  He  then  had 
%  61  in  the  bank.  How  much  money  had  he  in  the  bank  before 
depositing  the  $17? 

6.  After  drawing  out  $35  from  a  savings  bank  a  boy 
had  left  $  7.45  in  the  bank.  How  much  money  had  he  in  the 
bank  before  drawing  out  the  $35  ? 

7.  After  gaining  7  lb.  a  girl  weighed  103  lb.  How  much 
did  she  weigh  before  gaining  the  7  lb.  ? 

8.  George  and  Frank  together  have  as  much  money  as 
Walter.  George  has  $2.15  and  Walter  has  $4.10.  How 
much  money  has  Frank  ? 

9.  A  man  owns  three  farms  amounting  together  to  240 
acres.  Two  of  the  farms  contain  80  acres  and  120  acres  re- 
spectively.    How  many  acres  are  there  in  the  third  farm  ? 

10.  A  house  and  lot  together  cost  $4500.  The  lot  cost 
$  1500.     Find  the  cost  of  the  house. 

11.  The  sum  of  two  numbers  is  238.  One  of  the  numbers 
is  79.     What  is  the  other  number  ? 


THE  EQUATION  ^87 

12.  The  sum  of  the  three  sides  of  a  triangle  is  24  in.  One 
of  the  sides  is  8  in.  and  another  is  9  in.  What  is  the  length 
of  the  third  side  ? 

13.  After  selling  40  sheep  a  farmer  had  236  sheep.  How 
many  sheep  had  he  before  selling  the  40  sheep  ? 

14.  Write  3  problems  similar  to  each  of  Probs.  1-13  and 
write  the  equation  for  each. 

359.  1.  If  ^  is  added  to  both  sides  of  the  equation 
7  —  aj'=  2,  the  result  will  be  7  =  2  +  «•  What  will  be  the 
result  if  x  is  added  to  both  sides  of  the  equation  10  —  a;  =  7  ? 

2.  If  X  is  added  to  both  sides  of  the  equation  14  =  25  —  a;, 
the  result  will  be  14  +  a;  =  25.  The  value  of  x  is  found  by- 
adding  to  both  sides  of  the  equation  some  number  that  will 
leave  x  by  itself  on  the  left  side. 

3.  Write  each  of  the  following  equations  so  that  x  will  be 
by  itself  on  the  left  side  of  the  equation.  First,  add  x  to  both 
sides  of  the  equation,  then  write  the  equation  so  that  the  side 
containing  x  will  be  on  the  left.  18  =  43  —  iK ;  21  =  72  —  a; ; 
60-0^  =  37;  33-aj  =  19;  54  =  62-a;;  68-ic  =  28. 

Write  each  of  the  following  statements  so  that  x  will  be 
by  itself  on  the  left  side  of  the  equation,  and  solve : 

4.  167-a;  =  100.  8.    74-a;  =  18. 

5.  a;-$36  =  |75.  9.    $ 45.75- a?  =  $30.50. 

6.  15  lb.  =  25  lb. -a.-.  10.    65.4 -a;  =  18.45. 

7.  78  ft.  =  135  ft.  -X.  11.    125  da.  -  a;  =  ^h  da. 
12.   Write  10  equations  and  find  the  value  of  x  in  each. 

360.  1.  The  sum  of  x  and  x  and  x,  or  3  times  x,  is  written 
Zx.  Write  the  sum  of  x  and  x.  Write  the  product  of  4  times 
x\  of  5  times  a. 

2.  If  X  is  4,  what  is  the  value  of  2  a;  ?  Compare  a;  =  4  and 
2aj  =  8.  What  must  both  terms  of  a;  =  4  be  multiplied  by  to 
give  2  a;  =  8?  Multiply  both  terms  of  a;  =  3  by  5.  Has  this 
changed  the  value  of  x  in  the  equation  ? 


288  APPENDIX 

3.  State  what  effect  multiplying  both  sides  of  the  equation 
by  the  same  number  has  upon  the  vahie  of  x  in  an  equation. 
Prove  the  truth  of  your  statement. 

4.  Divide  both  sides  of  the  equation  Go;  =  18  by  2;  by  3; 
by  6.    Has  this  changed  the  value  of  x  in  the  several  equations  ? 

5.  State  what  effect  dividing  both  sides  of  tlie  equation  by 
the  same  number  has  upon  the  value  of  x  in  an  equation. 
Prove  the  truth  of  your  statement. 

6.  If  2a;  +  4  =  21,  what  is  the  value  of  2a;  ?  of  a;  ?  of  3a;  ? 

7.  If  x  =  6,  what  is  the  value  of  7a;  ?  of  3a;  ?  of  5a;  ? 
Find   the  value   of  x  in  each  of  the  following  equations. 

Where  the  equation  shows  an  unknown  quantity  to  be  sub- 
tracted from  one  side  of  the  equation,  add  this  unknown 
quantity  to  both  sides  of  the  equation  ;  then  write  the  equation 
with  the  unknown  quantity  on  the  left  side  and  solve : 


8. 

2a; -45  =  69. 

13. 

146 -3a;  =  83. 

9. 

24  =  78 -2a;. 

14. 

$35.40  -  3  a;  =  f  10.95. 

10. 

45  ft. -4a;  =  13  ft. 

15. 

f.85  =  $.40-f-3a;. 

11. 

345 -4  a;  =135. 

16. 

$90-6a;=$48. 

12. 

240  A.  =  880  A. -4a;. 

17. 

24yr. -3a;=6yr. 

361.    Solve  each  of  the  following  without  using  x.      Then 
solve  each,  using  x. 

1.  If  3  times  a  certain  number,  plus  25,  is  55,  what  is  the 
number  ? 

2.  If  4  times  a  certain  number,  less  20,  is  40,  what  is  the 
number  ? 

3.  Mary  is  20  years  old.     This  is  2  years  more  than  twice 
Edna's  age.     What  is  Edna's  age  ? 

4.  Walter  has  $45.      This  is  $13  more  than  4  times  the 
amount  of  money  James  has.     How  much  money  has  James? 

5.  If  4  times  a  certain  number,  plus  3  times  that  number, 
is  28,  what  is  the  number  ?     (4a;  -|-  3a;  =  7a;.) 


THE   EQUATION  289 

6.  If  6  times  a  certain  number,  plus  4  times  that  number, 
is  160,  what  is  the  number  ? 

7.  If  4  times  a  certain  number  is  the  same  as  6  times  18, 
what  is  the  number  ? 

8.  A  man  bought  three  railroad  tickets,  each  costing  the 
same  amount,  and  paid  $1.50  for  bus  rides.  He  paid  out 
$  6.90  in  all.     Find  the  price  paid  for  each  ticket. 

9.  The  sum  of  two  numbers  is  48,  and  one  number  is  5 
times  the  other.  What  are  the  numbers?  (Let  x  and  5x 
represent  the  numbers.) 

10.  A  man  bought  two  carriages.  For  one  he  paid  twice 
what  he  paid  for  the  other.  Both  carriages  cost  him  $210. 
Find  the  cost  of  each. 

11.  Write  problems  similar  to  each  of  the  above,  and  state 
the  equation  for  each. 

12.  Draw  an  oblong  whose  length  is  twice  its  width.  Let  x 
represent  its  width.  What  will  represent  its  length  ?  its 
perimeter?  If  the  perimeter  of  the  oblong  is  30  in.,  how 
wide  is  it  ?     How  long  is  it  ? 

13.  Draw  two  lines,  one  of  which  is  3  times  the  length  of 
the  other.  If  the  sum  of  their  lengths  is  24  ft.,  how  long  is 
each  line  ? 

14.  Two  men  together  own  540  acres  of  land.  One  owns 
twice  as  much  as  the  other.     How  many  does  each  own  ? 

15.  A  man  offered  to  divide  $10  between  two  boys  in  pro- 
portion to  their  ages,  provided  the  boys  could  tell  how  much 
each  should  receive.  The  boys  were  12  years  and  13  years 
respectively.  After  solving  the  problem  the  boys  stated  that 
the  younger  should  receive  $4.50  and  the  older  $5.50.  Did 
they  solve  it  correctly  ?     If  not,  what  is  the  correct  answer  ? 

AK.  — 19 


290  APPENDIX 

16.  A  man  offered  some  boys  $  1.50  for  weeding  his  garden. 
The  boys  found  that  they  could  not  all  work  at  the  same  time, 
so  the  man  agreed  to  pay  each  boy  the  same  wages  per  hour 
for  the  work  done.  One  boy  worked  7  hours,  another  worked 
5  hours,  and  the  third  worked  3  liours.  How  much  of  the 
money  should  each  boy  receive  ? 

17.  A  man  wished  to  leave  $  3500  to  his  three  sons  so  that 
the  second  son  would  receive  twice  what  the  youngest  received 
and  the  eldest  would  receive  4  times  what  the  youngest  re- 
ceived.    How  much  should  each  son  receive  ? 

362.  1.  The  expression  f  is  used  to  denote  ^  of  x.  Write 
the  expression  that  denotes  J  of  a; ;  -J-  of  x;  -J  of  a; ;  ^  of  a; ;  -f 
of  X. 

2.  By  what  number  must  f  be  multiplied  to  make  x? 
By  multiplying  both  sides  of  the  equation  f  =  3  by  4,  the 
equation  is  changed  to  a;  =  12. 

3.  Multiply  f  by  the  number  that  will  give  x  as  the  re- 
sult. Multiply  ^  by  the  smallest  whole  number  that  will  give 
a  whole  number  of  a;'s  as  the  result. 

4.  What  is  the  smallest  number  that  both  sides  of  the 
equation  f  =  12  can  be  multiplied  by  to  leave  only  whole 
numbers  in  the  equation?  Multiplying  both  sides  of  an 
equation  by  some  number  that  will  leave  the  equation  without 
fractional  quantities  is  called  clearing  the  equation  of  fractions. 

5.  Clear  the  following  equations  of  fractions :  f  =  8 ; 
1=14;  ¥  =  12;  |t  =  36;  45=^;  60  =  VV ;  -¥--8  =  1; 
72-^^  =  56. 

6.  Clear  the  following  of  fractions  and  find  the  value  of  x : 

Solve  each  of  the  following  without  using  x.  Then  solve 
each,  using  x : 

7.  Divide  60  into  two  numbers  such  that  the  first  is  J  of 
the  second. 


THE   EQUATION  291 

8.  Separate  36  into  two  parts  whose  ratio  is  -|. 

9,  Divide  $2.10  into  two  amounts  whose  ratio  is  the  same 
as  the  ratio  of  15^  to  20 /. 

10.  If  ^  of  a  certain  number,  plus  |  of  it,  is  39,  what  is  the 
number  ? 

11.  Solve  :    6  times  8  =  12  times  x ;    4  times  9  =  6  times  x. 

1 9       Snl  VP  •     6   a;,     a   —  12.     «   —   42 

J.4S.     OOlVe  .    9-  —  -3  ,    -g-  —  T6  J    T  —  4  9"- 

13.  The  equation  J  =  ^  may  be  cleared  of  fractions  by  mul- 
tiplying both  sides  of  the  equation  by  the  least  common  mul- 
tiple of  the  denominators.  This  is  2  x.  The  equation  is  thus 
changed  to  the  form  12  =  a;. 

14.  Solve:  i  =  ii;  ,\  =  i;  V"=l«;   I  =  Il- 
ls. Solve:  w  =  J^;  M  =  ¥;  A=^;  ^=^- 

16.  Write  ten  exercises  similar  to  exercises  8-12  and  find 
the  value  of  x  in  each. 

363.   Proportion. 

Solve  each  without  using  x.     Solve  each,  using  x : 
1.    The  shadow  of  a  post  5  ft.  high  is  3  ft.  6   in.  long. 
How  high  is  a  telephone  pole  whose  shadow  is  28  ft.  long  ? 

a.  The  height  of  the  post  is  -^  times  the  length  of  its 
shadow.  The  height  of  the  telephone  post  is  -^  times  28 
ft.     Explain. 

b.  The  length  of  the  shadow  of  the  post  is  in  the  same  ratio 
to  the  height  of  the  post  as  the  length  of  the  shadow  of  the 
telephone  pole  is  to  the  height  of  the  pole.  The  equality  of 
these  ratios  may  be  expressed  thus : 

3^^28 
5        X 
Solve  to  find  the  value  of  x,  the  number  of  feet  in  the  height 
of  the  telephone  pole. 

c.  The  equality  of  the  two  ratios  may  be  expressed  thus : 
3.5  ft.  :  5  f t.  :  :  28  ft.  :  x,  which  is  read,  3.5  ft.  is  to  5  ft.  as 
28  ft.  is  to  X.     The  first  and  last  terms  (as  3.5  ft.  and  x)  of  a 


292  APPENDIX 

proportion  are  called  the  extremes,  and  the  two  middle  terms 
the  means.  Tlie  product  of  the  extremes  in  a  proportion  is 
always  equal  to  the  product  of  the  means.  Hence,  3.5  times  x 
=  5  times  2S,  or  3.5  a;  =  140.  Solve  to  find  the  value  of  x,  the 
number  of  feet  in  the  height  of  the  telephone  pole.  This 
method  of  solving  a  proportion  differs  only  inform  of  expres- 
sion from  the  method  (b)  given  on  p.  201. 

2.  How  high  is  a  tree  whose  shadow  is  34  ft.  6  in.,  if  the 
shadow  of  a  boy  whose  height  is  4  ft.  9  in.  is  3  ft.  3  in.  ? 

3.  If  the  distance  traveled  by  a  trail*  in  1  hr.  45  min.  is  80 
mi.,  how  long,  at  the  same  rate  of  speed,  xWU  it  take  the  train 
to  travel  475  mi.  ? 

4.  Find  by  the  method  used  in  solving  Probs.  1  and  2  the 
height  of  objects  near  the  schoolhouse. 

MEASUREMENT  OF  SURFACES  AND   SOLIDS 
364.   Areas  of  Surfaces. 

1.  Draw  a  vertical  line;  a  horizontal  line;  an  oblique  line. 

2.  Draw  a  line  perpendicular  to  another  line ;  parallel  to 
another  line. 

3.  Draw  a  right  angle ;  an  acute  angle ;  an  obtuse  angle. 

4.  Draw  a  rectangle.  Is  a  rectangle  a  parallelogram? 
Draw  a  parallelogram  that  is  not  a  rectangle. 

5.  How  many  dimensions  has  a  rectangle  ?  Is  a  rectangle 
a  quadrilateral  ?  Draw  a  quadrilateral  that  is  not  a  parallelo- 
gram. 

6.  State  how  the  area  of  a  parallelogram  is  found.  Find  the 
area  of  a  parallelogram  whose  base  is  20  ft.  and  whose  altitude 
is  18  ft. 

7.  A  quadrilateral  that  has  only  two  parallel  sides  is  called 
a  trapezoid. 

8.  State  how  the  area  of  a  trapezoid  is  found.  Draw  a 
trapezoid.     Assign  its  dimensions  and  find  its  area. 


MEASUREMENT   OF   SURFACES   AND   SOLIDS      293 

9.  TVTiat  is  a  triangle  ?  Draw  a  right  triangle ;  an  acute- 
angled  triangle ;  an  obtuse-angled  triangle. 

10.  State  how  the  area  of  a  triangle  is  found.  Draw  a  tri- 
angle.    Assign  its  dimensions  and  find  its  area. 

11.  Make  a  drawing  to  show  the  relation  of  the  area  of  a 
triangle  to  the  area  of  a  parallelogram  having  the  same  base 
and  altitude. 

12.  Draw  a  parallelogram.  Draw  its  diagonals.  Do  they 
cross  at  the  middle  of  the  parallelogram  ? 

13.  What  is  meant  by  the  perimeter  of  a  figure?  Find  the 
perimeter  of  your  schoolroom. 

14.  Draw  a  circle.  Draw  its  radius ;  its  diameter.  Point  to 
its  circumference. 

15.  State  how  the  circumference  of  a  circle  is  found  when 
the  length  of  its  radius  is  known.  State  how  the  diameter  of 
a  circle  is  found  when  the  length  of  its  circumference  is  known. 

16.  State  how  the  area  of  a  circle  is  found.  Assign  the 
necessary  dimensions  and  find  the  area  of  a  circle. 

17.  State  how  the  area  of  the  convex  surface  of  a  cylinder 
is  found.  Find  the  area  (including  the  ends)  of  a  cylinder  whose 
diameter  is  6  ft.  and  whose  length  is  8  ft. 

365.   Regular  Polygons. 

1.  Mention  a  surface  that  is  a  plane  surface.  A  plane  figure 
bounded  by  straight  sides  is  called  a  polygon.  A  polygon 
whose  sides  are  all  equal  and  whose  angles  are  all  equal  is 
called  a  regular  polygon. 


o  o 


Triangle  .  Square  Penta^'ou  Hexagon 

Rbgdlak  Pulygons 


294  APPENDIX 

2.   A  regular  polygon  of  three  sides  is  called  an  equilateral 
triangle;  of  four  sides,  a  square;  of  five  sides,  a  pentagon;  of 
six  sides,  a  hexagon ;  of  seven  sides,  a  heptagon ; 
^'^N.  of  eight  sides,  an  octagon.     Draw  an  octagon. 

^  C-'^        ^*    ^   straight   line    from    the   center   of  a 

\         f '       /    regular   polygon   to   any  vertex   is  called  its 
\        ;        /      radius  (r). 

4.    The  perpendicular  from  the  center  of  a 
regular  polygon  to  any  side  is  called  its  apothem  (a). 

5.  The  area  of  a  regular  polygon  is  the  sum  of  the  areas  of 

the  triangles  formed  by  its  radii  and  sides. 
The  apothem  is  the  altitude  of  each  of  the 
triangles,  and  the  perimeter  is  the  sum  of 
the  bases  of  the  triangles.     Hence, 

The  area  of  a  regular  polygon  is  equal  to  one 
half  the  product  of  its  perimeter  and  apothem, 

6.  Draw  a  pentagon.  Assign  its  dimensions  and  find  its 
area. 

7.  Draw  a  hexagon.  Assign  its  dimensions  and  find  its 
area. 

8.  Draw  an  octagon.  Assign  its  dimensions  and  find  its 
area. 

9.  The  area  of  a  circle  is  one  half  the  product  of  its  radius 
and  circumference.  Compare  the  method  of  finding  the  area 
of  a  regular  polygon  with  this  method  of  finding  the  area  of  a 
circle. 

366.    Solids. 

1.  How  many  dimensions  has  a  plane  surface?  Name 
them. 

2.  How  many  dimensions  has  a  solid  ?     Name  them. 

3.  What  name  is  given  to  a  solid  whose  faces  are  all  rec- 
tangles ?  to  a  solid  whose  faces  are  equal  squares? 


MEASUREMENT   OF   SURFACES   AND  SOLIDS      295 

4.  What  name  is  given  to  a  solid  whose  ends  are  triangles 
and  whose  sides  are  rectangles  ? 

5.  State  how  the  volume  of  a  prism  is  found.  Draw  a 
prism.     Assign  its  dimensions  and  find  its  volume. 

6.  Name  solids  that  are  rectangular  prisms. 

7.  State  how  the  volume  of  a  cylinder  is  found.  Draw  a 
cylinder.  Assign  its  dimensions  and  find  its  volume.  Find 
its  area,  including  the  ends. 

367.    Pyramids  and  Cones. 

1.  A  solid  whose  base  is  a  polygon  and  whose  faces  are 
triangles  meeting  at  a  point  (vertex)  is  called 
a  pyramid. 

2.  The  area  of  the  surface  of  a  pyramid  is 
the  sum  of  the  areas  of  the  triangular  faces. 

3.  The  perpendicular  distance  from  the 
base  to  the  vertex  of  a  pyramid  is  called  its 
altitude  (vb). 

4.  The  altitude  of  one  of  the  triangular  faces  of  a  pyramid 
is  called  its  slant  height  (vs). 

5.  Construct  a  pyramid  of  cardboard.  Which  is  the 
greater,  the  altitude  of  a  pyramid  or  its  slant  height  ?  The 
apothem  of  a  polygon  forming  the  base  of  a  pyramid  may  be 
regarded  as  the  base  of  a  right  triangle  (bs),  the  altitude  as  the 
other  leg  (vb),  and  the  slant  height  as  the  hypotenuse  (vs). 
How  may  the  altitude  be  found  when  the  apothem  of  the  base 
and  the  slant  height  are  given  ? 

6.  Draw  a  regular  polygon.  Draw  its  radius  and  the 
apothem  of  an  adjacent  side.  The  figure  formed  by  the  radius, 
apothem,  and  one  half  of  the  adjacent  side  is  what  kind  of  a 
triangle  ?  if  the  radius  and  side  of  a  regular  polygon  are 
given,  how  tnay  the  apothem  be  found  ? 


296 


APPENDIX 


7.  If  the  altitude  of  a  pyramid,  the  radius 
of  its  base,  and  the  adjacent  side  are  given, 
how  may  the  slant  height  be  found  ? 

8.  A  solid  whose  base  is  a  circle  and 
which  tapers  to  a  point  called  the  vertex  or 
apex,  is  called  a  cone. 

9.  A  cone  may  be  regarded  as  a  pyramid 
whose  surface  is  an  infinite  number  of  narrow  triangles.  Its 
altitude  and  slant  height  correspond  to  the  altitude  and  slant 
height  of  a  pyramid. 

The  area  of  the  surface  of  a  pyramid  or  a  cone  is  equal  to  one 
half  the  product  of  its  slant  height  and  the  perimeter  of  its  base. 


Prism 


Pyramid 


Cylinder 


Cone 


10.  The  volume  of  a  pyramid  is  equal  to  one  third  the 
volume  of  a  prism  of  the  same  base  and  altitude,  and  the 
volume  of  a  cone  is  equal  to  one  third  the  volume  of  a  cylinder 
of  the  same  base  and  altitude.     Hence, 

TJie  volume  of  a  pyramid  or  a  cone  is  equal  to  one  third  the 
product  of  its  altitude  and  the  area  of  its  base. 

11.  A  cj^indrical  granite  stone  3  ft.  in  diameter  and  4  ft.  in 
height  was  cut  down  into  a  cone  of  the  same  base  and  altitude. 
What  part  of  the  stone  was  cut  away  ? 

368.   Spheres. 

TJie  area  of  the  surface  of  a  sphere  is  four  times  the  area  of  a 
great  circle  (irr')  of  the  sphere. 

1.   As  (2?-)',  or  4  r^  is  equal  to  d^,  4  irr'^  is  equal' to  tk?. 


PUBLIC   LANDS  297 

77i«  area  of  tlie  surface  of  a  sphere  ia  equal  to  the  square  of  the 
dmineter  x  vy  or  trd^, 

2.  Whicli  is  the  greater  and  lio"w  much,  the  area  of  a  cube  whose 
side  is  1  ft.  or  the  area  of  a  sphere  whose  diameter  is  1  ft.  ? 

3.  A  sphere  may  be  divided  into  an  infinite  number  of 
figures  that  are  essentially  pyramids.  The  combined  volume 
of  these  pyramids  is  the  volume 

of  the  sphere.  The  convex  sur- 
face of  the  sphere  may  be  re- 
garded as  the  sum  of  the  bases 
of  the  pyramids  and  the  radius 
of  the  sphere  as  the  altitude  of 
the  pyramids.     Hence, 

Tlie  volume  of  a  sphere  is  equal  to  one  third  the  product  of  its 
radius  and  its  convex  surface,  or  |  ttt^  (|  of  r  x  4  trr^). 

4.  As  d^,  or  (2iry  is  equal  to  8r^,  ^irr^  is  equal  to  ^ird^. 
Hence, 

To  find  the  volume  of  a  sphere,  multiply  the  cube  of  its  diameter 
by. 5236  a  of  ^). 

5.  The  earth  is  how  many  times  the  size  of  the  moon,  if  the 
diameter  of  the  earth  is  8000  mi.  and  the  diameter  of  the 
moon  is  2000  mi.? 

Volumes  of  spheres  are  to  each  other  as  the  cubes  of  their 
like  dimensions.  The  ratio  of  the  earth  and  moon  is  8^  (8000^) 
to  2^  (2000^),  or  4:^  to  1^. 

MEASUREMENT  OF  PUBLIC  LANDS 

369.  1.  At  the  time  the  colonial  settlements  were  made,  no 
uniform  system  of  measuring  lands  was  used.  Generally,  each 
settler  was  permitted  to  occupy  whatever  lands  he  wished,  and 
the  boundary  lines  were  often  designated  by  such  convenient 
natural  objects  as  rocks,  streams,  trees,  hilltops,  etc.  Later 
these  boundaries  were  recorded  as  the  legal  "  metes  and 
bounds ''  of  their  several  possessions.     These  tracts  of  land 


298 


APPENDIX 


were  often  so  irregular  in  shape  as  to  make  it  difficult  to  fix 
their  exact  boundaries  and  to  determine  their  exact  areas. 

2.  Shortly  after  the  close  of  the  Kevolutionary  War,  the 
Continental  Congress  appointed  a  committee,  of  which  Thomas 
Jefferson  was  chairman,  to  draw  up  some  plan  for  the  survey 
of  public  lands.  This  committee  reported  a  plan  which,  after 
being  slightly  amended,  was  adopted  by  Congress  in  1785,  and 
thus  became  the  government  system  of  measuring  public  lands. 

3.  In  accordance  with  this  system,  all  public  lands,  except 
"waste  and  useless  lands,"  have  been  laid  out  in  tracts  6  miles 
square  called  townships.  The  exact  location  of  each  township 
is  determined  by  north  and  south  lines  called  principal  meri- 
dians, and  by  east  and  west  lines  called  base  lines. 

Study  the  following  diagram : 


Standard. 


Base. 


Uj 
6^ 


B 


T.  5n. 
_  Parol lef 

T.  4  N. 


T.  3  N. 


T.  2  N. 


T.  I  N. 
—  Line 

T  I  S. 


T  2  S. 


4.  In  surveying  a  tract  of  land,  a  prominent  point  that  is 
easily  identified  and  is  visible  for  some  distance  is  established 
astronomically,  and  is  known  as  the  initial  (beginning)  point. 
In  the  figure,  the  initial  point  is  at  0. 


PUBLIC   LANDS  299 

5.  A  line  extending  north,  or  south,  or  both  north  and  south, 
from  the  initial  point  is  taken  as  a  principal  meridian.  The 
principal  meridian  is  the  true  meridian  at  the  initial  point. 
Locate  the  principal  meridian  in  the  figure. 

6.  A  line  extending  either  east  or  west,  or  both  east  and 
west,  through  the  initial  point,  or  a  line  perpendicular  to  the 
principal  meridian,  is  taken  as  a  base  line.  The  base  line  is 
always  a  true  parallel  of  latitude.  Locate  the  base  line  in 
the  figure. 

7.  East  and  west  lines  6  miles  apart,  called  town  lines,  are 
run  parallel  to  the  base  line,  and  north  and  south  meridian 
lines  6  miles  apart,  called  range  lines.  These  lines  divide  the 
tract  into  townships  6  miles  square.  Point  to  the  township 
lines  in  the  figure.  How  far  apart  are  these  lines  ?  Point  to 
the  range  lines.     How  far  apart  are  these  lines  ? 

8.  Point  to  a  township  in  the  first  tier  of  townships  north 
of  the  base  line.  Point  to  a  township  in  the  second  tier  of 
townships  north  of  the  base  line.  Point  to  a  township  in  the 
first  tier  of  townships  south  of  the  base  line. 

9.  A  township  in  the  third  tier  of  townships  north  of  a  base 
line  is  said  to  be  in  township  3,  north  (T.  3  N.).  A  township 
in  the  second  tier  of  townships  south  of  a  base  line  is  said  to 
be  in  township  2,  south  (T.  2  S.). 

10.  Point  to  the  first  north  and  south  row  of  townships,  east 
of  the  principal  meridian.  These  townships  are  said  to  be  in 
range  1,  east  (K.  1  E.).  Point  to  a  township  in  range  3,  east-, 
in  range  2,  west. 

11.  The  township  marked  A  is  numbered  township  4  north, 
range  2  east  (T.  4  N.,  R.  2  E.).  Describe  the  location  of  town- 
ships B,  C,  D,  E,  and  O.  Write  the  description  of  each,  using 
abbreviations. 

12.  Locate  in  the  figure  each  of  the  following  described  town- 
ships :  T.  2N.,  R.  3E.;  T.  4N.,  R.  5E.;  T.  IK,  R.  IW.; 
T.2S.,R.4E.;  T.1S.,R.3W.;  T.4K,R.2W.;  T.2S.,R.2E. 


300  APPENDIX 

la  Draw  a  diagram  showing  a  principal  meridian,  a  base 
line,  and  townships  and  ranges  as  in  the  figure  on  p.  298.  In 
your  diagram,  locate  the  following :  T.  4  S.,  K.  1  E. ;  T.  6  K., 
R.  5  W.;  T.  6K,  R.  6  E. 

14.  Locate  on  a  map  a  principal  meridian  and  a  base  line 
from  which  ranges  and  townships  in  your  state  are  numbered, 
if  the  land  has  been  measured  by  this  system.*  Give  the 
number  of  the  township  in  which  you  live.  Can  you  tell  the 
width  of  the  state  in  which  you  live  from  the  number  of  town- 
ships along  the  base  line?  Is  there  any  similarity  between 
the  method  of  locating  townships  by  means  of  principal  me- 
ridians and  base  lines  and  the  method  of  locating  places  on  the 
earth's  surface  by  means  of  degrees  of  longitude  and  latitude  ? 

15.  The  lands  of  Florida,  Alabama,  Mississippi,  of  the  states 
west  of  Pennsylvania  and  north  of  the  Ohio  River,  and  of  all 
states  west  of  the  Mississippi  River,  except  Texas,  have  been 
surveyed  in  the  manner  described.  Can  you  tell  from  your 
study  of  United  States  History  why  the  lands  of  the  other 
states  were  not  surveyed  in  this  manner  ? 

16.  The  initial  points  are  located  somewhat  arbitrarily. 
Sometimes  they  are  located  on  the  east  or  west  boundaries  of 
states,  at  other  times  they  are  located  at  the  junction  of  rivers, 
or  on  the  summits  of  elevations.  They  are  at  irregular  intervals 
apart.  Consequently,  the  land  in  a  single  state  may  be  meas- 
ured from  more  than  one  principal  meridian,  or  a  single  me- 
ridian may  be  used  for  measuring  the  land  in  several  states. 
Much  care  is  taken  to  preserve  the  exact  location  of  all  initial 
points. 

"An  initial  point  sliould  have  a  conspicuous  location,  visible  from 
distant  points  on  lines ;  it  should  be  perpetuated  by  an  indestructible 
monument,  preferably  a  copper  bolt  firmly  set  in  a  rock  ledge  ;  and  it 
should  be  witnessed  by  rock  bearings,  without  relying  on  anything 
perishable  like  wood."    Manual  of  Surveying  Instructions,  1902. 

•  Unmounted  land  maps  of  the  various  states  may  be  purchased  from 
the  Department  of  Interior,  Washington,  for  a  few  cents. 


PUBLIC   LANDS 


301 


17.  As  the  lines  that  bound  the  ranges  on  the  east  and  west 
are  true  meridians,  they  converge  as  they  extend  north  from  a 
base  line.  As  a  result,  townships  are  not  true  squares.  To 
correct  the  effect  of  the  convergeucy  of  the  meridians,  standard 
parallels  (formerly  called  correction  lines)  are  established  at 
regular  intervals  (now  24  miles  apart)  from  the  base  line, 
and  new  meridians  are  established  6  miles  apart  on  the  stand- 
ard parallels.  Guide  meridians  are  also  established  at  inter- 
vals (now  24  miles  apart),  east 
and  west  of  the  principal  meridi- 
ans, to  correct  inaccuracies  in 
measurement. 


370.    Townships. 

1.  A  township  is  a  tract  of  land 
6  miles  square.  It  contains  36 
square  miles  of  land.  A  square 
mile  of  land  is  called  a  section. 
The  sections  of  a  township  are 
numbered  as  shown  in  the  dia- 
gram. The  sections  of  a  town- 
ship are  numbered,  respectively,  beginning  with  number  1 
in  the  northeast  section  and  numbering  west  and  east  alter- 
nately.    Draw  a  township  and  number  the  sections. 

2.  Section  16  of  each  township  in  the  state  was  granted  by 
Congress  to  the  states  for  educational  purposes.  This  section 
is  therefore  commonly  known  as  the  school  section,  and  all 
moneys  derived  from  the  rent  or  sale  of  these  sections  is 
placed  in  the  public  school  fund  of  the  state.  States  that 
have  been  organized  since  1852  have  been  granted  two  sections 
in  each  township  for  the  support  of  public  schools,  sections  16 
and  36. 

Owing  to  the  convergency  of  the  meridians  that  bound  the 
townships  on  the  east  and  west,  a  township  is  never  exactly  6 
miles  from  east  to  west,  and  does  not  therefore  contain  36  full 


6 

5 

4 

3 

2 

1 

7 

8 

9 

10 

n 

12 

18 

17 

B 

15 

14 

13 

19 

20 

2/ 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

■ 

A  Township  divided  into 
Sections. 


302 


APPENDIX 


sections  of  G40  acres  each.  The  survey  of  the  sections  in  each 
township  is  begun  in  the  southeast  corner  of  the  township,  and 
all  sections  except  those  along  the  western  and  northern  bound- 
aries of  the  township  are  1  mile  square,  and  contain  640  acres 
each.  All  excess  or  deficiency  is  added  to  or  deducted  from 
the  sections  along  the  western  and  northern  boundaries  of  the 
township.  These  sections  generally  contain  less  than  640 
acres.  The  sections  along  the  western  boundary  of  a  township 
often  contain  less  than  630  acres.  Section  6  is  frequently  re- 
duced to  about  620  acres. 

371.   Sections. 

1.  A  section  is  subdivided  into  quarter  sections,  and  these  are 
again  subdivided  into  quarters,  etc.,  as  shown  in  the  diagram. 

2.  The  part  of  the  section  marked  A  is  described  as  the  west 
one  half  (W.  ^)  of  the  section,  and  contains  320  acres.    The  part 

marked  5  is  described  as  the  south- 
east quarter  of  the  section,  and 
contains  160  acres.  C  is  the  west 
one  half  of  the  northeast  one  fourth 
of  the  section  (W.  |  of  N.E.  J). 
How  many  acres  does  it  contain  ? 

The  part  marked  F  is  described 
as  the  S.E.  \  of  the  S.E.  \  of  the 
N.E.  J  of  the  section.  How  many 
acres  does  it  contain  ? 

3.  Describe  the  part  marked  O 
and  tell  bow  many  acres  it  contains. 

4.  Describe  the  part  marked  E  and  tell  how  many  acres  it 
contains. 

5.  Describe  the  part  marked  D  and  tell  how  many  acres  it 
contains. 

6.  Draw  a  section  and  subdivide  it  to  show  the  following 
and  give  the  number  of  acres  in  each : 

7.  N.W.  \  of  the  N.E.  J.  8.   S.W.  J  of  the  N.W.  J. 


D 


.\G 
\F 


A  Sbction  Subdivided. 


PUBLIC   LANDS  303 


9.    E.  I  of  the  S.W.  J.  10.   W.  I  of  the  N.W.  f 

11.  S.E.  -1-  of  the  S.W.  i  of  the  KE.  \. 

12.  S.  i  of  the  S.E.  i  of  the  N.E.  i. 

372.  Using  the  scale  1  in,  =  1  mi.,  draw  a  plot  to  represent 
a  township,  say  T.  6  'N.,  K.  4  E.  ;  locate  and  find  the  area  of 
each  of  the  following : 

1.  E.  i  of  the  S.E.  ^  of  Sec.  9,  T.  6  N.,  E.  4  E. 

2.  N.  W.  1  of  the  S.E.  J  of  Sec.  22,  T.  6  N.,  K.  4  E. 

3.  S.E.  1  of  the  S.W.  i  of  the  S.E.  Jof  Sec.  32,  T.  6  N.,  R.4  E. 

4.  E.  i  of  the  S.W.  i  of  the  N.E.  i  of  Sec.  24,  T.  6  N.,  R. 
4  E.,  which  is  a  farm  owned  by  Mr.  Thomas. 

5.  S.E.  1  of  the  K.W.  ^  of  Sec.  18,  T.  6  N.,  R.  4  E.,  which 
is  the  description  of  a  piece  of  property  on  which  Mr.  White 
pays  taxes. 

373.  Review. 

1.  The  unit  of  land  measure  is  the  township,  which  is  theo- 
retically 6  miles  square.  The  word  town  is  commonly  used 
for  township. 

2.  What  are  initial  points  ?  principal  meridians  ?  base 
lines  ? 

3.  What  is  a  range  ?  How  many  sections  are  there  in  a 
township  ?     How  are  they  numbered  ? 

4.  How  many  acres  are  there  in  a  section  ?  in  a  quarter 
section  ? 

5.  Public  lands  are  generally  sold  in  sections,  half  sections, 
quarter  sections,  and  in  half  quarter  sections.  What  part  of  a 
section  is  80  acres  ?  40  acres  ?  20  acres  ? 

6.  How  many  acres  are  there  in  a  full  township  ?  in  a  full 
section  ? 

7.  What  are  standard  parallels,  or  correction  lines  ? 

8.  Which  are  the  school  sections  ?     Why  are  they  so  called  ? 


804  APPENDIX 

9.  Cau  you  tell  from  your  study  of  United  States  history 
why  some  uniform  system  of  surveying  public  lands  was 
necessary  soon  after  the  close  of  the  Revolutionary  War  ? 

10.  "What  sections  generally  contain  less  than  640  acres  ? 
Why? 

11.  Locate  the  principal  meridian  and  the  base  line  used  in 
measuring  the  land  in  which  your  schoolhouse  is  located. 

12.  A  new  standard  parallel  is  located  at  intervals  of  24 
miles  north  or  south  of  the  base  line,  and  a  new  guide  meridian 
is  located  at  intervals  of  24  miles  east  and  west  of  a  principal 
meridian.     Make  a  diagram  showing  these  lines. 

THE   METRIC   SYSTEM   OF  WEIGHTS   AND   MEASURES 

374.  1.  The  system  of  denominate  units  of  measure  in  com- 
mon use  in  the  United  States  is  practically  the  same  as  that  in 
use  in  Great  Britain,  with  the  exception  of  the  units  used  in 
measuring  value.  Nearly  all  the  other  civilized  nations  use  a 
decimal  system  of  denominate  numbers,  called  the  metric 
system.  The  metric  system  has  been  legalized  by  the  United 
States  and  Great  Britain,  and  has  been  adopted  as  the  sys- 
tem for  use  in  the  Philippines  and  Porto  Rico.  It  is  exten- 
sively used  in  scientific  work. 

2.  A  little  more  than  a  century  ago  the  French  government 
invited  the  nations  of  the  world  to  a  conference  to  consider  an 
international  system  of  weights  and  measures.  Later,  the 
French  government  appointed  a  committee  to  devise  a  conven- 
ient system  of  denominate  units.  The  committee  originated 
what  is  known  as  the  Metric  System  of  Weights  and  Meas- 
ures. The  metric  system  includes  measures  of  length,  surface, 
capacity,  volume,  and  weight.  The  primary  unit  of  linear 
measure  is  the  meter  The  primary  unit  of  each  of  the  other 
measures  is  based  upon  the  meter. 

3.  One  ten-millionth  part  of  the  distance  from  the  e(]na<^or 
to  the  North  Pole,  measured  on  the  meridian  of  Paris,  \\</» 


METRIC   SYSTEM  305 

selected  as  the  primary  unit  of  linear  measure.  This  unit  is 
called  the  meter.  Meter  is  the  French  word  for  measure.  The 
meter  is  a  little  longer  than  the  yard.  As  it  is  based  upon  a 
measurement  of  the  earth's  polar  circumference,  the  meter 
is  a  fixed  natural  unit.* 

4.  An  International  Bureau  of  Weights  and  Measures  has 
been  established  in  Paris,  and  is  now  supported  by  the  contri- 
butions of  more  than  twenty  nations.  A  standard  meter, 
made  from  an  alloy  of  platinum  and  iridium,  is  carefully  pre- 
served by  this  bureau.  All  the  nations  of  the  world  have  been 
furnished  with  duplicates  of  this  standard  meter.  These 
duplicates  are  made  of  the  most  durable  and  least  expansible 
metals  known.  The  United  States  Bureau  of  Standards  has 
fixed  the  legal  equivalent  of  the  meter  as  39.37  inches. 

5.  The  metric  system  is  a  decimal  system.  Units  larger 
than  the  primary  units  are  10  times  the  primary  units,  100 
times  the  primary  units,  and  1000  times  the  primary  units; 
units  smaller  than  the  primary  units  are  -^^  the  primary  units, 
Y^^  the  primary  units,  and  y^Vir  ^^®  primary  units.  Units  of 
any  given  denomination  are  therefore  reduced  to  units  of  a 
larger  denomination  by  dividing  by  10,  by  100,  and  by  1000 ; 
and  units  are  reduced  to  units  of  a  smaller  denomination  by 
multiplying  by  10,  by  100,  and  by  1000.  Quantities  are  not 
generally  expressed  in  terms  of  two  or  more  units,  but  in  some 
single  unit,  parts  of  the  unit  being  expressed  as  a  decimal  of  the 
unit,  as  6.35  meters. 

6.  Names  of  units  larger  than  the  primary  units  are  formed 
by  prefixing  to  the  names  of  the  primary  units  prefixes  de- 
rived from  the  Greek  words  meaning  ten,  one  hundred,  and 
one  thousand,  etc.;  and  names  of  units  smaller  than  the 
primary  units  are  formed  by  prefixing  to  the  names  of  the 
primary  units  prefixes  derived  from  the  Latin  words  meaning 
ten,  one  hundred,  and  one  thousand,  as  follows ; 

*  Subsequent  calculations  have  shown  that  the  meter  is  not  exactly  a 
ten-millionth  part  of  the  distance  from  the  equator  to  the  North  Pole. 

AR.  —  20 


306  APPENDIX 

Greek  Prefixes 
dekay  meaning  10;  dekameter,  meaning  10  meters. 
hekto,  meaning  100;  hektometer,  meaning  100  meters. 
kilo,  meaning  1000;  kilometer,  meaning  1000  meters. 
myria,  meaning  10,000 ;  myriameter,  meaning  10,000  meters. 
The  prefixes  deka  and  hekto  are  sometimes  written  dcca  and  hecto. 

Latin  Prefixes 
deci,  meaning  10 ;  decimeter,  meaning  .1  meter. 
ceyiti,  meaning  100 ;  centimeter,  meaning  .01  meter. 
milli,  meaning  1000 ;  millimeter,  meaning  .001  meter. 

Very  small  linear  measurements  are  expressed  in  mikrons.  Mikron 
is  a  Greek  word  meaning  small. 

375.   Measures  of  Length. 

In  the  following  exercises  use  a  meter  stick  on  which  the 
centimeters  and  millimeters  are  marked  off.  Practice  drawing 
these  units  until  you  can  estimate  their  lengths  quite  accu- 
rately.    Test  all  estimates  by  actual  measurements. 

1.  Draw  on  the  blackboard  a  line  1  meter  long ;  2  meters 
long ;  3  meters  long. 

2.  Fix  two  points  on  the  floor  1  meter  apart;  2  meters 
apart ;  3  meters  apart ;  4  meters  apart. 

3.  Estimate  the  length,  width,  and  height  of  your  school- 
room in  meters. 

4.  Measure  the  length  of  a  blackboard  in  meters.  Express 
fractional  parts  as  a  decimal  of  a  meter,  thus :  if  the  black- 
board is  4  meters  and  12  centimeters  long,  its  length  may  be 
stated  as  4.12  meters. 

5.  Estimate  the  length  and  width  of  the  school  yard  in 
metevQ. 

'^.  Draw  a  line  1  decimeter  in  length.  Name  some  object 
in  the  schoolroom  that  is  one  decimeter  in  lengtli,  width,  or 
thickness. 


METRIC   SYSTEM  307 

7.  Draw  a  line  1  centimeter  in  length,  2  centimeters  in 
length,  3  centimeters  in  length. 

8.  Measure  the  length  and  width  of  this  book  in  centi- 
meters.    Express  fractional  parts  as  a  decimal  of  a  centimeter. 

9.  Measure  the  thickness  of  this  book  in  millimeters. 
How  many  millimeters  make  a  centimeter  ?  a  decimeter  ?  a 
meter  ? 

10.  A  kilometer  is  1000  meters.  It  is  equivalent  to  about  | 
of  a  mile.  Select  some  place  that  is  about  1  kilometer  from 
the  schoolhouse. 

11.  Using  rulers  on  which  the  units  are  marked  off,  com- 
pare the  millimeter  with  -^^  of  an  inch. 

12.  Which  is  the  longer,  a  centimeter  or  an  inch  ? 

376.   Reduction  of  Linear  Units. 

1.  A  meter  is  how  many  decimeters  ?  how  many  centi- 
meters ?  how  many  millimeters  ? 

2.  67  centimeters  may  be  expressed  as  a  decimal  of  a 
meter,  thus :  .67  meter.  Express  as  meters :  34  centimeters, 
15  centimeters,  76  centimeters. 

3.  A  decimeter  is  what  part  of  a  meter  ?  4  decimeters  may 
be  expressed  as  a  decimal  of  a  meter,  thus  :  .4  meter.  Ex- 
press as  meters :  3  meters  and  4  decimeters ;  7  meters  and  32 
centimeters  ;  9  meters,  2  decimeters,  and  4  centimeters. 

4.  Write  a  millimeter  as  a  decimal  of  a  meter.  Write  8 
millimeters  as  a  decimal  of  a  meter.  Write  3  centimeters  and 
8  millimeters  as  a  decimal  of  a  meter. 

5.  Write  2  kilometers  as  meters.  Write  2  kilometers  and 
430  meters  as  meters.  Write  as  meters  :  24.5  kilometers;  4.25 
kilometers. 

6.  Reduce  to  meters:  304  centimeters;  2.467  kilometers; 
245.376  kilometers ;  30  centimeters. 


308  APPENDIX 

377.  Table  of  Measures  of  Length. 

The  following  is  the  complete  table  of  linear  measure.  The 
units  most  commonly  used  are  the  millimeter,  centimeter, 
meter,  and  kilometer. 

1000  mikrons  (/a)  =  I  millimeter  (mm.) 

10  mm.  =  1  centimeter  (cm.) 

10  cm.  =  1  decimeter  (dm.) 

10  dm.  =  1  meter  (m.) 

10  m.  =  1  dekameter  (Dm.) 

10  Dm.  =  1  hektometer  (Hm.) 

10  Hm.  =  1  kilometer  (Km.) 

10  Km.  =  1  myriameter 

Abbreviations  of  the  names  of  the  units  that  are  multiples  of  the  pri- 
mary unit  are  written  with  capital  letters  to  distinguish  them  from  the 
abbreviations  of  the  names  of  the  units  that  are  parts-  of  the  primary 
unit. 

378.  Measures  of  Surface. 

1.  Draw  on  the  blackboard  a  square  whose  side  is  1  meter 
in  length.     This  is  called  a  square  meter. 

2.  Divide  a  square  meter  into  square  decimeters.  How 
many  square  decimeters  are  there  in  a  square  meter  ? 

3.  Divide  a  square  decimeter  into  square  centimeters. 
How  many  square  centimeters  are  there  in  a  square  decimeter? 

4.  How  many  square  millimeters  are  there  in  a  square 
centimeter  ? 

5.  How  many  square  centimeters  are  there  in  a  square 
meter  ? 

6.  In  what  square  unit  should  you  express  the  area  of  the 
surface  of  the  cover  of  this  book  ?  of  the  floor  of  your  school- 
room ? 

7.  Draw  on  the  school  grounds  a  square  whose  side  is  10 
meters.  This  is  called  an  are.  It  is  the  primary  unit  of  land 
measure. 

The  are  is  equivalent  to  119.6  square  yards. 


METRIC   SYSTEM  309 

8.  A  square  whose  side  is  100  meters  is  called  a  hektare. 

9.  A  square  whose  side  is  1  kilometer  is  called  a  square 
kilometer. 

The  area  of  gardens,  etc.  is  usually  given  in  ares ;  of  fields,  etc.  in 
hektares  ;  and  of  countries,  etc.  in  square  kilometers. 

10.  Estimate  the  number  of  square  meters  in  tho  surface  of 
the  floor  of  your  schoolroom.     Test  your  estimate. 

11.  Estimate  the  number  of  ares  in  the  school  yard.  Test 
your  estimate. 

12.  How  long  is  the  side  of  a  hektare  ?  of  a  square  kilo- 
meter ? 

The  hektare  is  nearly  2|  acres. 

379.  Table  of  Measures  of  Surface. 

100  square  millimeters  (qmm.)  =1  square  centimeter  (qcm.) 

100  qcm.  =  1  square  decimeter  (qdm.) 

100  qdm.  =  1  square  meter  (qm.) 

100  qm.        ■  =1  square  dekameter  (qDm. ) 

100  qDm.  =  1  square  hektometer  (qHm.) 

100  qHm.  =  1  square  kilometer  (qKm.) 

380.  Table  of  Land  Measure. 

100  centares  (ca.)  =  1  are  (a.) 

100  a.,  =1  hektare  (Ha.) 

381.  Measures  of  Volume. 

1.  From  a  piece  of  cardboard  construct  a  cube  whose  edges 
are  each  1  decimeter.    This  is  called  a  cubic  decimeter. 

2.  How  many  cubic  decimeters  are  there  in  1  cubic  meter  ? 

3.  From  a  piece  of  cardboard  construct  a  cubic  centimeter. 
Kstimate  the  capacity  of  a  crayon  box  in  cubic  centimeters. 

4.  Estimate  the  number  of  cubic  meters  of  air  in  your 
pchoolroom.  Using  a  meter  stick,  make  an  approximate  test 
(•f  your  estimate. 


810  APPENDIX 

5.  The  primary  unit  of  volume  is  the  cubic  meter. 
The  cubic  meter  is  equivalent  to  1.308  cubic  yards. 

6.  The  primary  unit  of  wood  measure  is  the  stere,  which  is 
a  cubic  meter. 

382.  Table  of  Measures  of  Volume. 

1000  cubic  millimeters  (cu.  mm.)  =  1  cubic  centimeter  (cu.  cm.) 
1000  cu.  cm.  r=  1  cubic  decimeter  (cu.  dm.) 

1000  cu.  dm.  =  1  cubic  meter  (cu.  m.) 

Units  higher  than  the  cubic  meter  are  seldom  used. 

383.  Measures  of  Capacity. 

1.  The  primary  unit  of  capacity  for  both  liquid  and  dry 
measure  is  the  liter,  which  contains  1  cubic  decimeter.  Using 
the  measures,  compare  the  capacity  of  a  liter  and  a  quart. 

The  liter  is  equivalent  to  1.0567  liquid  quarts  or  .908  dry  quart. 

2.  How  many  cubic  centimeters  are  equivalent  to  1  liter? 

3.  100  liters  are  1  hektoliter.  The  liter  is  used  to  measure 
comparatively  small  quantities ;  the  hektoliter  is  used  to  meas- 
ure grain,  produce,  etc.,  in  large  quantities. 

The  hektoliter  is  equivalent  to  2.8377  bushels. 

4.  Mention  some  things  that  are  bought  or  sold  by  the 
quart,  dry  measure ;  by  the  quart  or  gallon,  liquid  measure. 
Where  the  metric  system  is  used,  these  are  bought  and  sold  by 
the  liter,  or  by  the  hektoliter  if  the  quantities  are  large. 

5.  How  many  liters  of  water  will  a  tank  hold  whose  inside 
dimensions  are  3.45  m.  by  80  cm.  by  60  cm.? 

846  X  80  X  60 


1000 


-,  number  of  liters  in  the  tank.    Explain. 


6.    Find  the  capacity  in  liters  of  a  cylindrical  tank  whose 
diameter  is  2.85  m.  and  whose  altitude  is  3.68  m. 


METRIC   SYSTEM  311 

384.  Table  cf  Measures  of  Capacity. 

10  milliliters  =  1  centiliter  (cl.) 
10  cl.  =  1  deciliter  (dl.) 

10  dl.  =  1  liter  (1.) 

10  1.  =1  dekaliter  (Dl.) 

10  Dl.  =lhektoliter  (HI.) 

385.  Measures  of  Weight. 

1.  The  primary  unit  of  weight  is  the  gram,  which  is  the 
weight  of  1  cu.  cm.  of  pure  water  at  its  greatest  density. 

2.  Heft  a  gram  weight.  How  many  grams  does  a  liter  of 
pure  water  at  its  greatest  density  weigh  ? 

3.  The  weight  of  1000  cubic  centimeters  of  water  (a  liter)  is 
called  a  kilogram,  or  a  kilo.     Heft  a  kilogram  weight. 

A  kilogram  is  equivalent  to  2.2046  pounds  avoirdupois.  How  many- 
grams  are  equivalent  to  an  ounce  avoirdupois  ? 

4.  The  gram  is  used  in  weighing  precious  metals,  medicines, 
etc. ;  the  kilogram  in  weighing  meat,  groceries,  etc.  Express 
your  weight  in  kilograms,  calling  2.2  pounds  1  kilogram. 

5.  100  kilograms  are  1  metric  quintal,  and  1000  kilograms 
1  metric  ton. 

A  metric  ton  is  equivalent  to  2206  pounds  or  1.1023  tons. 

6.  Express  as  grams:  2.125  Kg. ;  3.4  Kg.  Express  as  kilo- 
grams :  245  g. ;  28  g. ;  362  M.  T. ;  4.25  M.  T. ;  4  Kg.  72  g. 

386.  Table  of  Measures  of  Weight. 

10  milligrams  (mg.)  =  1  centigram  (eg.) 

10  eg.  =1  decigram  (dg.) 

10  dg.  =  1  gram  (g.) 

10  g.  =1  dekagram  (Dg.) 

10  Dg.  =  1  hektogram  (Hg.) 

10  Hg.  =  1  kilogram  (Kg.) 

10  Kg.  =  1  myriagram  (Mg.) 

100  Kg.  =  1  metric  quintal  (Q.) 

1000  Kg.  =  1  metric  ton  (M.  T.) 


312  APPENDIX 

387.   Equivalents  of  Metric  Units. 

The  following  equivalents  are  given  for  comparison  and  for 
reference : 


Mktric  to  Common 

Common  to  Mktkk- 

1  m. 

=  39.37  in.,  or  1.0936  yd. 

1  yd.                =  .9144  m. 

IKm. 

=  .62137  mi. 

1  mi.               =1.60935  Km. 

1  sq.  m 

.  =  1.196  sq.  yd. 

1  sq.  yd.          =  .836  sq.  m. 

1  Ha. 

=  2.471  A. 

1  A.                =  .4047  Ha. 

1  cu.  m 

.  =  1.308  cu.  yd. 

1  cu.  yd.         =  .765  cu.  m. 

11. 

=  .908  qt.  (dry) 

Iqt.  (dry)     =1.1012  1. 

11. 

=  1.0567  qt.  (liquid) 

1  qt.  (liquid)  =  .94636  1. 

IHl. 

=  2.8377  bu. 

1  bu.               =  .35239  HI. 

Ig. 

=  15.43  gr.  (troy) 

loz.  (troy)   =  31.10348  g. 

IKg. 

=  32.1507  oz.  (troy) 

1  lb.  (av.)      =  .45359  Kg. 

IKg. 

=  2.2046  lb.  (av.) 

1  M.  T. 

=  1.1023  T. 

1  T.                =  .90718  M.  T. 

TABLES   OF  DENOMINATE   MEASURES 

(For  Reference) 

388. 

Measures  of  Time. 

60  seconds  =  1  minute 

365  days    =1  year 

60  minutes  =  1  hour 

366  days    =  1  leap  y^ar 

24  hours       =  1  day 

10  years    =  I  decade 

7  days        =  1  week 

100  years  =  1  century 

1.  The  day  is  the  primary  unit  of  time  measure.  It  is  the 
time  taken  by  the  earth  to  make  one  rotation  on  its  axis.  Is 
it  a  natural  or  an  artificial  unit  ?  The  earth  revolves  around 
the  sun  in  365  days  5  hours  48  minutes  46  seconds  (nearly 
365\  days).     This  period  is  the  solar  (sun)  year. 

2.  As  the  exact  period  taken  for  the  earth  to  make  a  revolu- 
tion around  the  sun  is  a  little  less  than  365 J^  days,  an  extra  day 
(Feb.  29)  is  added  to  the  common  year  once  in  four  years  (leap 
year),  except  in  centennial  years  not  exactly  divisible  by  400. 

3.  Centennial  years  dioisihh  by  400  and  other  years  divisible 
by  4  «^<?  leap  years. 

^Vas  1700  a  leap  year  ?    Will  2000  be  a  leap  year  ? 


TABLES   OF  DENOMINATE  MEASURES  313 

4.  More  than  four  thousand  years  ago  the  Chaldeans,  a  people 
living  in  the  valley  of  the  Euphrates,  calculated  the  length  of 
the  year  to  be  360  days.  They  believed  that  the  sun  traveled 
around  the  earth  in  a  circle  in  this  period.  They  therefore 
divided  the  circular  path  of  the  sun  into  360  equal  parts, 
called  degrees  —  one  for  the  part  traversed  each  day.  Hence 
there  are  360  degrees  in  a  circle.  They  observed  twelve  clusters 
of  stars  (constellations)  in  the  zone  in  the  heavens  (zodiac)  in 
which  the  paths  of  the  sun  and  planets  lie,  and  the  occurrence 
of  twelve  full  moons  in  successive  parts  of  the  zodiac  each 
year.  They  therefore  divided  the  course  of  the  sun  into 
twelve  equal  parts,  one  for  each  constellation.  Hence  there 
are  twelve  months  in  a  year.  The  exact  length  of  the  lunar 
month  is  29.53059  days.  The  Chaldeans  divided  the  day  into 
twelve  "  double  hours.'^  The  number  60  was  used  by  them  as 
a  unit,  and  they  therefore  divided  the  hour  and  the  degree 
into  60  minutes ;  and  the  minute  into  60  seconds. 

5.  Seven  days  were  made  to  constitute  a  unit  of  time  meas- 
ure (week),  either  in  accordance  with  the  Mosaic  law  or  from 
the  fact  that  seven  planets  were  known  to  the  ancients.  The 
days  of  the  week  were  originally  named  after  seven  heavenly 
bodies.  The  English  names  of  the  days  of  the  week  are  derived 
from  the  Saxons,  a  Germanic  people  who  invaded  and  con- 
quered England  in  the  fifth  and  sixth  centuries.  The  Saxons 
borrowed  the  week  from  some  eastern  nation  and  substituted 
the  names  of  their  own  divinities  for  those  of  the  Grecian 
deities. 

Names  of  the  Days  of  the  Week 


Latin 

Saxon 

ENGLISn 

Dies  Solis  (Sun) 

Sun's  day 

Sunday 

Dies  Lunae  (Moon) 

Moon's  day 

Monday 

Dies  Martis  (Mars) 

Tiw's  day 

Tuesday 

Dies  Mercurii  (Mercury) 

Woden's  day 

Wednesday 

Dies  Jovis  (Jupiter) 

"Thor's  day 

Thursday 

Dies  Veneris  (Venus) 

Frlga's  day 

Friday 

Dies  Saturni  (Saturn) 

Seterne*s  day 

Saturday 

314  APPENDIX 

6.  Until  the  time  of  Julius  Caesar  (46  b.c.)  the  calendar  was 
in  almost  constant  state  of  confusion,  owing  to  the  fact  that 
the  number  of  days  allowed  for  a  year  was  more  or  less  than 
the  actual  number  of  days  taken  for  one  revolution  of  the 
earth  iu  its  orbit.  As  a  result  of  this  error,  in  the  time  of 
Julius  Caesar  the  winter  months  had  been  carried  back  into 
autumn,  and  the  autumn  months  into  summer.  To  correct  the 
error,  Caesar  decreed  that  90  days  should  be  added  to  the  year 
to  restore  the  time  of  the  vernal  equinox,  and  that  the  year 
should  consist  of  365J  days.  He  ordered  that  the  common 
year  should  thereafter  consist  of  365  days  and  that  every 
fourth  year  should  consist  of  366  days.  The  extra  day  was 
added  to  February,  which  at  that  time  had  29  days.  This 
arrangement  is  known  as  the  Julian  Calendar,  or  Old  Style. 
The  month  of  July  was  named  after  Julius  Caesar. 

7.  Augustus  Caesar  ordered  that  the  month  following  that 
which  bore  the  name  of  Julius  (July)  should  be  named  after 
himself;  and  in  order  that  the  month  bearing  his  name  should 
have  as  many  days  as  the  month  bearing  the  name  of  Julius, 
he  ordered  that  one  day  be  taken  from  February  and  added  to 
the  month  which  should  bear  his  name.  Hence  the  eighth 
month  is  named  August  and  consists  of  as  many  days  as  July. 

8.  The  year  established  by  the  Julian  Calendar  (365J  days) 
was  .00778  of  a  day  longer  than  the  actual  time  taken  for  one 
revolution  of  the  earth  in  its  orbit.  This  error  had  amounted 
to  10  days  by  1582,  when  Pope  Gregory  XIII  undertook  the 
correction  of  the  calendar.  To  adjust  the  time  of  the  vernal 
equinox,  Pope  Gregory  ordered  that  ten  days  be  skipped,  from 
October  5th  to  the  15th,  and  that  only  centennial  years  that  are 
exactly  divisible  by  400  and  other  years  that  are  exactly  divis- 
ible by  4  be  made  leap  years.  This  arrangement  is  known 
as  the  Gregorian  Calendar,  or  New  Style,  and  is  the  one  in 
common  use.  Kussia  still  follows  the  Julian  or  Old  Style. 
The  error  in  the  Gregorian  Calendar  will  amount  to  one  day  in 
about  5000  years. 


TABLES   OF   DENOMINATE   MEASURES  315 

389.    Measures  of  Length. 

12  inches  =  1  foot 
3  feet      =  1  yard 
16|  feet  (5^  yd.)  =  1  rod 
320  rods  =  1  mile 
1  mile  =  1760  yards  =  5280  feet 

1.  The  yard  is  the  primary  unit  of  length.  All  the  other 
units  of  lengtli  are  derived  from  it. 

2.  A  furlong  is  \  mile.     It  is  little  used  at  the  present  time. 

3.  A  hand,  used  in  measuring  the  height  of  horses  at  the 
shoulder,  is  4  inches. 

4.  A  fathom,  used  in  measuring  the  depth  of  the  sea,  is  6  feet. 

5.  A  knot,  or  nautical  mile,  used  in  measuring  distances  at  sea, 
is  6080.27  feet,  or  approximately  1.15  (about  1^)  miles.  The 
speed  of  vessels  is  expressed  in  knots.  A  vessel  that  travels 
18  knots  an  hour  travels  about  21  miles  an  hour  (18  mi.  plus  | 
of  18  mi.). 

6.  For  the  supposed  origin  of  the  inch,  foot,  fathom,  etc., 
consult  a  dictionary  or  an  encyclopedia. 


390.   Measures  of  Surface. 

144  square  inches  =  1  square  foot 
9  square  feet      =1  square  yard 
30J  square  yards  =  1  square  rod 
160  square  rods     =  1  acre 
640  acres  =  1  square  mile 


1.  A  square  acre  is  208.71  +  feet  on  a  side. 

2.  A  tract  of  land  1  mile  square  is  called  a  section.     A  town- 
ship is  a  tract  of  land  6  miles  square  and  consists  of  36  sections. 

3.  100  square  feet  of  flooring,  roofing,  or  slating  is  called  a 
square. 

391.   Measures  of  Volume. 

1728  cubic  inches  =  1  cubic  foot 
27  cubic  feet     =  1  cubic  yard 


316  APPENDIX 

1.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high, 
or  128  cubic  feet  of  wood,  is  called  a  cord.  For  the  origin  of  the 
name,  consult  the  dictionary.  Stonework  is  sometimes  meas- 
ured by  the  cord. 

2.  In  measuring  stonework,  a  pile  of  stone  16^  feet  long,  1^ 
feet  wide,  and  1  foot  high,  or  24|  cubic  feet  of  stone,  is  called 
a  perch. 

392.  Surveyors*  Measures  of  Length. 

100  links  (l.)=l  chain  (ch.) 
80  chains      =  1  mile 

The  chain  in  common  use  is  called  Gunter's  chain.  It  is 
4  rods,  or  66  feet  long.  A  link  is  .66  foot.  Links  are  written 
as  hundredths  of  a  chain,  thus :  30  chains  45  links  is  written 
30.45  chains. 

393.  Surveyors'  Measures  of  Surface. 

10  square  chains  =  1  acre 

640  acres  =  1  square  mile 

Square  chains  are  reduced  to  acres  by  moving  the  decimal 
roint  one  place  toward  the  left.     Explain. 

394.  Avoirdupois  Weight. 

16  ounces  =  1  pound 
100  pounds  =  1  hundredweight 
2000  pounds  =  1  ton 

1.  The  English  ton,  known  in  the  United  States  as  the  long 
ton,  is  2240  pounds.  It  is  used  in  United  States  custom- 
houses and  in  weighing  coal  and  mineral  products  at  the  mines 
and  sometimes  in  retailing  coal. 

2.  The  smallest  unit  of  weight  is  the  grain.  A  i)ound  avoir- 
dupois is  7000  grains.  Consult  a  dictionary  for  an  explanation 
of  the  origin  of  the  name. 


TABLES  OF   DENOMINATE   MEASURES  317 

3.  Grains,  vegetables,  etc.,  are  commonly  sold  by  weight  or 
measure.  The  weight  of  1  bushel  of  the  most  common  of  these 
articles  is  as  follows : 

wheat  -  60  lb.  oats  =  32  lb. 

beans  =  60  lb.  barley  =  48  lb. 

peas  =  60  lb.  sweet  potatoes  =  55  lb. 

clover  seed  =  60  lb.  rye  =  56  lb. 

Irish  potatoes  =  60  lb.  shelled  corn  =  56  lb. 

395.  Troy  Weight. 

Troy  weight  is  used  in  weighing  precious  metals. 

24  grains  =  1  pennyweight 

20  pennyweights  =  1  ounce 
12  ounces  =  1  pound 

A  pound  troy  is  5760  grains.     It  is  f^f  ^  pound  avoirdupois. 

Precious  stones  and  pearls  are  weighed  by  the  carat.  A 
carat  equals  3^  grains  troy.  The  term  carat  is  used  also  to 
express  the  proportion  of  gold  in  an  alloy.  It  then  signifies  a 
twenty-fourth  part.  Thus,  gold  that  is  18  carats  fine  is  ^|,  or 
I  pure  gold. 

396.  Apothecaries'  Weight. 

Consult  a  dictionary  for  the  meaning  of  the  word  apothecary. 
This  system  of  weights  is  used  to  some  extent  in  filling  pre- 
scriptions. The  pound,  ounce,  and  grain  are  the  same  as  in 
troy  weight,  but  the  ounce  is  subdivided  differently. 

20  grains  (gr.)  =  1  scruple  .  .  .     sc.  or  3 

3  scruples  =  1  dram  .  .  .     dr.  or  3 

8  drains  =  1  ounce  .  .  .    oz.  or  ^ 

12  ounces  =  1  pound  .  .  .lb.  or  lb 

397.  Apothecaries'  Liquid  Measures. 

60  drops  (gtt.)  or  minims  (TTL)  =  I  fl»i<l  dram     .     .     .  /3 

8  fluid  drams  =  1  fluid  ounce     .     •     •  /  5 

16  fluid  ounces  =  1  pint O. 

8  pints  =  1  gallon Cong. 


318  APPENDIX 

39o.    Liquid  Measures. 

4  gills      =  1  pint 
2  pints    =  1  quart 
4  quarts  =  1  gallon 

QuaH  means  one  fourth.  A  quart  is  one  fourth  of  a  gallon. 
A  gallon  is  231  cubic  inches.  A  gallon  of  water  weighs  about 
8^  pounds.  A  cubic  foot  of  water  (about  7^  gal.)  weighs  about 
62^  pounds.  In  measuring  the  capacity  of  cisterns,  etc.,  31^ 
gallons  are  called  a  barrel. 

399.  Dry  Measures. 

This  system  is  but  little  used  in  some  parts  of  the  United 
States.     Where  it   is  not  used,  articles  are  usually  sold  by 

W^^Slit-  2  pints  =  1  quart 

8  quarts  =  1  peck 
4  pecks  =  1  bushel 

The  dry  quart  contains  67.20  cubic  inches,  the  fluid  quart 
57.75  cubic  inches.  A  bushel  contains  2150.42  cubic  inches. 
The  standard  bushel  in  the  United  States  is  the  Winchester 
bushel.  It  is  the  volume  of  a  cylinder  18. V  inches  in  internal 
diameter  and  8  inches  in  depth. 

400.  Measures  of  Angles  and  Arcs. 

GO  seconds  (")  =  1  minute  (') 

60  minutes         =  1  degree  (°) 
360  degrees  =  4  right  angles,  or  1  circumference 

90°  of  angle         =  1  right  angle  ;  90"^  of  arc  =  1  quadrant 

For  an  explanation  of  the  origin  of  3G0  degrees  in  a  circum- 
ference, etc.,  see  Measures  of  Time,  p.  313. 

401.  Counting  Table. 

2  units  =  1  pair  20  units   =  1  score 

12  units  =  1  dozen  12  dozen  =  1  gross 

12  gross  =  1  great  gross 


TABLES   OF   DENOMINATE  MEASURES  319 

402.  Measures  of  Value  —  United  States  Money. 

10  mills  =  1  cent  10  dimes    =  1  dollar 

10  cents  =  1  dime  10  dollars  =  1  eagle 

The  standard  unit  of  value  is  the  gold  dollar.  A  gold  dol- 
lar (no  longer  coined)  contains  23.22  grains  of  pure  gold  and 
2.58  grains  of  alloy.  A  silver  dollar  contains  371.25  grains  of 
pure  silver  and  41.25  grains  of  alloy.  The  symbol  for  dollar 
..is  $,  which  is  taken  from  U.S. 

The  coins  of  the  United  States  are  bronze,  1^ ;  nickel,  5  ^ ; 
silver,  10^,  25^,  50/,  $1;  and  gold  $2i,  $5,  $10,  and  $20. 
The  mill  is  not  coined.  These  are  coined  at  mints  located  in 
Philadelphia,  New  Orleans,  Denver,  and  San  Francisco. 

The  paper  currency  is  issued  in  the  denominations  of  $1, 
$  2,  $5,  $  10,  $20,  $50,  $100,  $500,  and  $  1000.  Paper  cur- 
rency consists  of  bank  notes,  silver  certificates,  and  gold  cer- 
tificates. Examine  some  paper  currency.  The  provision  made 
for  the  redemption  of  each  piece  of  paper  currency  is  printed 
on  each  bill. 

Paper  currency  issued  by  national  banks  is  commonly  called  bank  notes. 
Their  payment  is  guaranteed  by  deposits  of  government  bonds  with  the 
national  government. 

403.  Values  of  Common  Coins. 


Country 

Monetary  Unit 

Valttk  in  Terms 

OF  U.S.  Gold 

Dollar 

KOTIQH 

Equivale: 

Austria-Hungary 

Crown 

$    .203 

J$.20 

Britisli     Possessions, 

N.  A.  (except  New- 

foundland) 

Dollar  ($) 

$  1.000 

$1.00 

France 

Franc  (F.) 

$    .193 

$   .20 

German  Empire 

Mark  (M.) 

$   .238 

$   .25 

Great  Britain 

Pound  Sterling  (£) 

^  4.8661 

$5.00 

Italy 

Lira  (L.) 

$    .193 

%   .20 

Japan 

Yen  (y.) 

$    .498 

$   .50 

Mexico 

Peso 

$    .498 

$   .60 

Philippine  Islands 

Peso 

1    .500 

$    .50 

Russia 

Ruble 

$   .515 

$   .60 

320 


APPENDIX 


404.   Table  of  Compound  Interest. 

Aiiwunt  of^l,  at  various  rates,  interest  compounded  annually. 


Yeaus 

1% 

lV,7o 

2% 

2%% 

3% 

3Va% 

1 

1.010000 

1.015000 

1.020000 

1.025000 

1.030000 

1.0.35000  1 

2 

1.020100 

1.030225 

1.040400 

1.060625 

1.060900 

1.071225 

3 

1.030301 

1.045678 

1.061208 

1.076891 

1.092727 

1.108718 

4 

1.040604 

1.061364 

1.082432 

1.103813 

1.126509 

1.147523 

5 

1.051010 

1.077284 

1.104081 

1.131408 

1.159274 

1.187686 

6 

1.061520 

1.093443 

1.126162 

1.169693 

1.194052 

1.229265 

7 

1.072135 

1.109845 

1.148686 

1.188686 

1.229874 

1.272279 

8 

1.082857 

1.126403 

1.171659 

1.218403 

1.266770 

1.316809 

9 

1.093685 

1.143390 

1.195093 

1.248863 

1.304773 

1.362897 

10 

1.104622 

1.160541 

1.218994 

1.280085 

1.343916 

1.410599 

11 

1.115668 

1.177949 

1.243374 

1.312087 

1.384234 

1.459970 

12 

1.126825 

1.195618 

1.268242 

]  .344889 

1.425761 

1.511069 

13 

1.138003 

1.213552 

1.293607 

1.378511 

1.468634 

1.66.3956 

14 

1.149474 

1.231756 

1.310479 

1.412974 

1.612690 

1.618695 

15 

1.160969 

1.250232 

1.345868 

1.448298 

1.567967 

1.676349 

16 

1.172579 

1.268986 

1.372786 

1.484506 

1.604706 

1  '33986 

17 

1.184304 

1.288020 

1.400241 

1.521618 

1.652848 

1.794676 

18 

1.196148 

1.307.341 

1.428246 

1.56i)659 

1.702433 

1.867489 

19 

1.208109 

1.326951 

1.456811 

1.598660 

1.753506 

1.922601 

20 

1.220190 

1.346855 

1.486947 

1.638616 

1.806111 

1.989789 

Years 

4% 

4V2% 

6% 

G% 

7% 

8% 

1 

1.040000 

1.045000 

1.060000 

1.060000 

1.070000 

1.080000 

2 

1.081600 

1.092025 

1.102500 

1.123600 

1.144900 

1.16(5400 

3 

1.124864 

1.141166 

1.167626 

1.191016 

1.225043 

1.259712 

4 

1.169859 

1.192519 

1.215506 

1.262477 

1.3107{>6 

1.3(50489 

5 

1.216653 

1.246182 

1.276282 

1.338226 

1.402562 

1.469328 

6 

1.265319 

1.302260 

1.340096 

1.418519 

1.500730 

1.586874 

7 

1.315932 

1.360862 

1.407100 

1.503630 

1.606782 

1.713824 

8 

1.368569 

1.422101 

1.477456 

1.693848 

1.718186 

1.850930 

9 

1.423312 

1.486095 

1.651328 

1.689479 

1.838459 

1.991HXI5 

10 

1.480244 

1.652969 

1.628895 

1.790848 

1.967151 

2.168925 

11 

1.539454 

1.622863 

1.710339 

1.898299 

2.104852 

2.331639 

12 

1.601032 

1.095881 

1.796856 

2.012197 

2.252192 

2.618170 

13 

1.665074 

1.772196 

1.886649 

2.132928 

2.409845 

2.719(524 

14 

1.731676 

1.851946 

1.979932 

2.260904 

2.578634 

2.937194 

15 

1.800944 

1.935282 

2.078928 

2.396668 

2.75iK)32 

3.172169 

16 

1.872981 

?.022370 

2.182875 

2.640352 

2.952164 

S.  426943 

17 

1.947901 

2.113377 

2.292018 

2.692773 

3.168816 

3.700018 

18 

2.026817 

2.208479 

2.406619 

2.864339 

3.379932 

3.9J)6020 

19- 

2.106849 

2.307860 

2.526950 

3.026600 

3.616528 

4.316701 

20 

2.191123 

2.411714 

2.653298 

3.207136 

3.869684 

4.660967 

INDEX 


Abstract  number,  16. 
Accident  insurance,  278. 
Accounts,  42,  43,  274. 
Acute  angle,  79,  221,  229. 
Acute-angled  triangle,  221,  229. 
Ad  valorem  duty,  198,  273.    • 
Addend,  IG. 
Addition,  of  denominate  numbers  77. 

effractions,  97-99,  108-112. 

of  integers  and  decimals,  16-19. 
Additive  method  of  subtraction,  22. 
Aliquot  parts,  147,  208. 
Altitude,  221,  226,  295. 
Amount,  in  addition,  16. 

in  interest,  208. 
Angle  measure,  237,  318. 
Angles,  79,  221,  229,  318. 
Apothecaries'  measures,  317. 
Apothem,  294. 
Appendix,  256-320. 
Approximate  ansvrers,  28. 
Arabic  numerals,  10. 
Arc,  237,  318. 
Are,  309. 

Areas,  78,  80,  225-232,  235,  292. 
Assessed  valuation,  198,  270. 
Assessors,  198,  270. 
Austrian  method  of  subtraction,  22. 
Avoirdupois  weight,  316,  317. 

Bank,  of  deposit,  276. 

savings,  274. 
Bank  accounts,  274. 
Bank  discount,  218. 
Bank  notes,  276. 
Banking,  274-278. 
Base,  221,  226. 
Base  line,  299. 
BiUs,  42,  43. 

and  receipts,  44. 
Board  foot,  160. 
Bonds,  261. 
Broker,  192,  258,  264. 
Brokerage,  258,  262-264. 


Calendar,  812-314. 

Cancellation,  105. 

Cancellation  method,  213. 

Capacity,  measures  of,  233,  234,  310,  811. 

Capital,  256. 

Carat,  317. 

Cash  discount,  204. 

Certificate  of  deposit,  277. 

Check,  277. 

Cipher,  10. 

Circle,  231,  232,  237. 

area  of,  232. 
Circular  measure,  237,  318. 
Circumference,  222,  231,  237. 
City  lot,  134. 
Clearing  house,  278. 
Coins,  value  of,  319. 
Collector,  of  the  port,  273. 

of  taxes,  270. 
Commercial  discount,  204. 
Commission,  192,  193,  262-264. 
Common  divisor,  factor,  or  measure,  104. 
Common  multiple,  106. 
Common  stock,  258. 
Composite  number,  87. 
Compound  denominate  numbers,  72,  77. 
Compound  interest,  217. 

table  of,  320. 
Concrete  number,  16. 
Cone,  295,  296. 
Consumer,  262. 
Corporation,  256. 
Corporation  bond,  261. 
Correspondence  bank,  278. 
Counting  measure,  318. 
Coupon  bond,  261. 
Credit,  creditor,  42. 
Cube  (rectangular  prism),  233. 
Cube  of  numbers,  244. 
Cube  root,  246. 

Cubic  measure,  84,  85,  233,  309,  310,  815. 
Customhouse,  198,  273. 
Customs  and  duties,  198,  202,  203,  271-278. 
Cylinder,  222,  234,  235. 

—  21  321 


322 


INDEX 


Dates,  difference  between,  168,  267. 

Days  of  grace,  215. 

Debit,  debtor,  42. 

Decimal  point,  9. 

Decimal  system,  7. 

Decimals,  addition  of,  18. 

division  of,  C8-70. 

multiplication  of,  89-41. 

notation  and  numeration  of,  15. 

reduction  of,  156-158. 

subtraction  of,  25. 
Degree,  237. 
Denominate  numbers,  72,  70-86. 

tables  of,  312-319. 
Denominator,  93,  109. 
Deposit,  bank  of,  276. 
Deposit  slip,  274. 
Diagonal,  226. 
Diameter,  222,  231. 
Difference,  20. 
Direct  taxes,  269. 
Discount,  171. 

bank,  218. 

cash,  204. 

trade  or  commercial,  204,  265. 

true,  220. 
Dividend,  in  division,  45. 

in  insurance,  280. 

in  stocks,  257. 
Divisibility  tests,  87. 
Division,  effractions,  117, 118,  124-131. 

of  integers  and  decimals,  45-70. 
Divisor,  45. 
Draft,  bank,  278. 
Dry  measure,  318. 
Duties,  198,  202,  208,  271-278. 

Endowment  policy,  279. 
Equation,  283-292. 
Equator,  238. 
Equivalents,  312. 
Even  number,  87. 
Evolution,  246. 
Exact  interest,  268. 
Exponent,  244. 
Extremes,  292. 

Face  of  note,  207. 
Factor,  45,  87,  104. 
Factoring,  89,  104. 

roots  found  by,  246,  247. 
Farm  problems,  86. 
Fire  insurance,  195. 
Flooring,  161. 
Foreign  money,  819. 
Forms,  221-248. 
Fraction  deflued,  98. 


Fractional  unit,  93. 
Fractions,  90-165, 

addition  and  subtraction  of,  97-99, 108-112. 

multiplication  and  division  of,  113-131. 

reduction  of,  96,  96,  100-105,  166-168. 

Gain  and  loss,  181-183. 

Government  expenses,  272. 

Gram,  311. 

Greatest  common  divisor,  factor,  ormeasure, 

104. 
Gregorian  calendar,  314. 

Health  insurance,  278. 
Heptagon,  294. 
Hexagon,  294. 
Horizontal  lines,  221, 228. 
Hypotenuse,  250. 

Improper  fraction,  94. 
Index  of  roots,  246. 
Indirect  taxes,  270. 
Insurance,  195,  196,  278-283. 
Integers,  defined,  9. 
Integers  and  decimals,  7-89. 
Interest,  by  aliquot  parts,  208. 

cancellation  method,  218. 

compound,  217,  320. 

defined,  167. 

exact,  268,  269. 

simple,  207-213. 

six  per  cent  method,  218. 

sixtv  dav  method,  212. 

tables,  267,  320. 
Internal  revenue,  198. 

Joint  note,  214. 

Land  measure,  297,  809. 

Latitude,  288. 

Law  of  commutation,  32. 

Least  common  denominator,  109. 

Least  common  multiple,  106. 

Life  insurance,  278-283. 

Life  policies,  279. 

Like  quantities,  16. 

Linear  measure,  76,  808,  816. 

Liquid  measure,  318. 

List  prices,  204. 

Liter,  310. 

Local  taxes,  198,  270. 

Long  division,  63-67. 

Long  measure,  76,  808,  816. 

Longitude  and  time,  240-242. 

Loss  and  profit,  181-188. 

Lowest  terms,  105. 

Lumber  measure,  160-162. 


INDEX 


323 


Maker  of  note,  214. 

Marine  insurance,  195. 

Market  reports,  264. 

Maturity  of  note,  215. 

Means,  292. 

Measurement,  division  by,  48. 

Measurements,     76-86,     160-165,     221-243, 

292-297. 
Measures,  312. 

Merchants'  rule  for  partial  payments,  216. 
Meridian,  238,  299. 
Meter,  804. 
Meter  reading,  75. 
Metric  system,  304-312. 
Minuend,  20. 
Mixed  number,  94. 
Model  bill,  42. 
Multiple,  45,  106. 
Multiplicand,  31. 
Multiplication,  31. 

effractions,  113-116,  119-123. 

of  integers  and  decimals,  81—44. 
Multiplier,  31. 
Municipal  corporation,  261. 

Negotiable  note,  208. 
New  style  calendar,  314. 
Notation  and  numeration,  10-15. 
Note,  207,  208,  214. 
Number  relations,  145,  146. 
Numerals,  10. 
Numeration,  10-15. 
Numerator,  98. 

Oblique  angle,  79. 

Oblique  line,  221,  223. 

Obtuse  angle,  79,  221,  229. 

Obtuse-angled  triangle,  221,  229. 

Octagon,  294. 

Odd  number,  87. 

Old  style  calendar,  814. 

Par  of  stock,  257. 

Parallel  lines,  80,  221,  223. 

Parallelogram,  221,  227. 

Partial  payments,  216,  266. 

Partition,  48,  51. 

Payee,  207. 

Payer,  207. 

Pentagon,  294. 

Percentage,  166-220. 

Perfect  square,  245. 

Periods,  10. 

Perpendicular  lines,  79,  221,  223. 

Personal  insurance,  278. 

Personal  property,  198,  270, 

Pi  {T),  231, 


Policy,  195,  279. 

Poll  tax,  270. 

Polygons,  regular,  293-294. 

Powers  and  roots,  244-251. 

Preferred  stock,  258. 

Premium,  on  policy,  195. 

stock  at,  257. 
Price  lists,  265. 
Prime  meridian,  238. 
Prime  number,  87. 
Principal,  207. 
Prism,  221,  222,  283-235. 
Proceeds,  218. 
Producers,  262. 
Product,  81. 
Profit  and  loss,  181-188. 
Promissory  note,  207,  214. 
Proper  fraction,  94. 
Property,  198,  269. 
Property  tax,  270. 
Proportion,  243,  291. 
Public  lands,  297-299. 
Pyramid,  295,  296. 

Quadrilateral,  221,  227. 
Quotient,  45. 

Radical,  246. 
Radius,  222,  231. 
Railway  time  table,  74. 
Range  lines,  299. 
Rate,  of  dividend,  257. 

of  interest,  215,  274. 

of  taxation,  198. 
Ratio,  54,  91,  145,  243. 
Real    estate    or    real    property,    198,    264, 

270. 
Receipts,  44. 
Reciprocals,  125. 
Rectangle,  80,  81,  221,  224,  252. 
Rectangular  solid,  84,  221,  224,  225,  233, 
Reduction  of  fractions,  100. 
Registered  bond,  261. 
Reviews,  71-75,    137-144,  150-155,  184-187, 

254,  255. 
Right  angle,  79,  221,  223,  237. 
Right-angled  triangle,  221,  226,  250,  251. 
Roman  notation,  13. 
Roots,  244-251. 

Savings  banks,  274. 
Scale  drawing,  132-136. 
Section,  301,  802. 
Shares  of  stock,  256. 
Shingling,  162. 
Short  methods,  88,  148. 
Similar  figures,  253. 


324 


INDEX 


Similar  fractions,  97. 

Similar  surfaces  and  solids,  253. 

Six  per  cent  method,  213. 

Sixty  day  method,  212. 

Slant  height,  295. 

Solar  year,  812. 

Solids,  84,  294-297. 

Specific  duty,  198,273. 

Sphere,  29C,  297. 

Square,  rectangle,  80,  224,  294, 

second  power,  244. 
Square  measure,  78,  315. 
Square  root,  247-250. 
Standard  time,  241. 
State  taxes,  198,  270. 
Stere,  310. 

Stock  quotations,  259, 
Stocks  and  bonds,  250-202, 
Subtraction,  of  denominate  numbers,  77. 

effractions,  97-99,  108-112. 

of  integers  and  decimals,  20-30. 
Subtrahend,  20. 
Sum,  10. 

Surface  measure,  78,  308,  809, 815. 
Surveyors'  measure,  81G. 

Tariff,  272. 

Tax  collector,  270. 

Tax  rate,  198. 


Taxes,  198-201,  269-271. 
Term  policy,  279. 
Terms  effraction,  94. 
Time  measure,  812,  318. 
Tontine  policy,  281. 
Town  lines,  299. 
Townships,  298,  801. 
Trade  discount,  204,  205,  265. 
Trapezoid,  228,  229,  292. 
Triangle,  221,  226,  229,  230,  294. 
Triangular  prisms,  222,  233. 
Troy  weight,  817. 


Unit,  7. 

Unit,  fractional,  93. 

Unit  of  measure,  7. 

United  States  money,  14,  25,  819. 

United  States  rule,  2CG. 

Unknown  quantity,  283. 

Usury,  215. 

Value,  table  of,  319. 

Vertex,  226. 

Vertical  line,  221. 

Volume,  233-235,  309,  810,  315,  816. 

Weight,  measures  of,  811,  816,  317. 


re  35874 


ivi55974        ^^ 


Mp: 

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